:: MATRTOP3 semantic presentation

REAL is non empty non trivial V36() V155() V156() V157() V161() set
NAT is non trivial ordinal V36() cardinal limit_cardinal V155() V156() V157() V158() V159() V160() V161() Element of bool REAL
bool REAL is non empty non trivial V36() set
COMPLEX is non empty non trivial V36() V155() V161() set
omega is non trivial ordinal V36() cardinal limit_cardinal V155() V156() V157() V158() V159() V160() V161() set
bool omega is non empty non trivial V36() set
K214() is TopStruct
the carrier of K214() is set
bool NAT is non empty non trivial V36() set
RAT is non empty non trivial V36() V155() V156() V157() V158() V161() set
INT is non empty non trivial V36() V155() V156() V157() V158() V159() V161() set
[:REAL,REAL:] is non empty non trivial Relation-like V36() complex-yielding ext-real-valued real-valued set
bool [:REAL,REAL:] is non empty non trivial V36() set
K375() is non empty strict multMagma
the carrier of K375() is non empty set
<REAL,+> is non empty strict unital Group-like associative commutative left-invertible right-invertible invertible left-cancelable right-cancelable V182() multMagma
K381() is non empty strict associative commutative left-cancelable right-cancelable V182() SubStr of <REAL,+>
<NAT,+> is non empty strict unital associative commutative left-cancelable right-cancelable V182() uniquely-decomposable SubStr of K381()
<REAL,*> is non empty strict unital associative commutative multMagma
<NAT,*> is non empty strict unital associative commutative uniquely-decomposable SubStr of <REAL,*>
{} is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
2 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
[:1,1:] is non empty Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
bool [:1,1:] is non empty V36() V40() set
[:[:1,1:],1:] is non empty Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
bool [:[:1,1:],1:] is non empty V36() V40() set
[:[:1,1:],REAL:] is non empty non trivial Relation-like V36() complex-yielding ext-real-valued real-valued set
bool [:[:1,1:],REAL:] is non empty non trivial V36() set
[:[:REAL,REAL:],REAL:] is non empty non trivial Relation-like V36() complex-yielding ext-real-valued real-valued set
bool [:[:REAL,REAL:],REAL:] is non empty non trivial V36() set
[:2,2:] is non empty Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
[:[:2,2:],REAL:] is non empty non trivial Relation-like V36() complex-yielding ext-real-valued real-valued set
bool [:[:2,2:],REAL:] is non empty non trivial V36() set
TOP-REAL 2 is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL 2) is non empty set
[:COMPLEX,COMPLEX:] is non empty non trivial Relation-like V36() complex-yielding set
bool [:COMPLEX,COMPLEX:] is non empty non trivial V36() set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial Relation-like V36() complex-yielding set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial V36() set
[:RAT,RAT:] is non empty non trivial Relation-like RAT -valued V36() complex-yielding ext-real-valued real-valued set
bool [:RAT,RAT:] is non empty non trivial V36() set
[:[:RAT,RAT:],RAT:] is non empty non trivial Relation-like RAT -valued V36() complex-yielding ext-real-valued real-valued set
bool [:[:RAT,RAT:],RAT:] is non empty non trivial V36() set
[:INT,INT:] is non empty non trivial Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued set
bool [:INT,INT:] is non empty non trivial V36() set
[:[:INT,INT:],INT:] is non empty non trivial Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued set
bool [:[:INT,INT:],INT:] is non empty non trivial V36() set
[:NAT,NAT:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
bool [:[:NAT,NAT:],NAT:] is non empty set
K610() is set
0 is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() V86() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of NAT
K74(0,1,2) is non empty V36() V155() V156() V157() V158() V159() V160() set
[:K74(0,1,2),K74(0,1,2):] is non empty Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
[:[:K74(0,1,2),K74(0,1,2):],K74(0,1,2):] is non empty Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
bool [:[:K74(0,1,2),K74(0,1,2):],K74(0,1,2):] is non empty V36() V40() set
bool [:K74(0,1,2),K74(0,1,2):] is non empty V36() V40() set
F_Real is non empty non degenerated non trivial right_complementable almost_left_invertible strict unital associative commutative right-distributive left-distributive right_unital well-unital V136() left_unital V189() V190() V191() doubleLoopStr
K560() is non empty Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K562() is non empty Relation-like [:REAL,REAL:] -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
doubleLoopStr(# REAL,K560(),K562(),1,0 #) is strict doubleLoopStr
the carrier of F_Real is non empty non trivial V155() V156() V157() set
{{},1} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
K886() is set
bool K886() is non empty set
K887() is Element of bool K886()
[:NAT,REAL:] is non trivial Relation-like V36() complex-yielding ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial V36() set
[:NAT,COMPLEX:] is non trivial Relation-like V36() complex-yielding set
bool [:NAT,COMPLEX:] is non empty non trivial V36() set
K400(NAT) is V187() set
the carrier of F_Real * is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
REAL * is non empty functional FinSequence-membered FinSequenceSet of REAL
3 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
REAL 0 is non empty functional FinSequence-membered FinSequenceSet of REAL
0 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
TOP-REAL 0 is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
0. (TOP-REAL 0) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal 0 -element V61( TOP-REAL 0) V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of the carrier of (TOP-REAL 0)
the carrier of (TOP-REAL 0) is non empty set
the ZeroF of (TOP-REAL 0) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal 0 -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of the carrier of (TOP-REAL 0)
{(0. (TOP-REAL 0))} is non empty trivial functional V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
- 1 is non empty V11() real ext-real non positive negative V85() set
Seg 1 is non empty trivial V16() V36() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
{1} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
Seg 2 is non empty V16() V36() 2 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
{1,2} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
[1,1] is set
{1,1} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{{1,1},{1}} is non empty V36() V40() set
[1,2] is set
{{1,2},{1}} is non empty V36() V40() set
[2,1] is set
{2,1} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{2} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{2,1},{2}} is non empty V36() V40() set
[2,2] is set
{2,2} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{{2,2},{2}} is non empty V36() V40() set
0. F_Real is V11() real ext-real V61( F_Real ) Element of the carrier of F_Real
the ZeroF of F_Real is V11() real ext-real Element of the carrier of F_Real
sin is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
cos is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:REAL,REAL:]
cos 0 is V11() real ext-real Element of REAL
sin 0 is V11() real ext-real Element of REAL
PI is V11() real ext-real Element of REAL
2 * PI is V11() real ext-real Element of REAL
PI / 2 is V11() real ext-real Element of REAL
2 " is non empty V11() real ext-real positive non negative set
PI * (2 ") is V11() real ext-real set
- (PI / 2) is V11() real ext-real Element of REAL
[.(- (PI / 2)),(PI / 2).] is V155() V156() V157() Element of bool REAL
K196(REAL,REAL,sin,[.(- (PI / 2)),(PI / 2).]) is V155() V156() V157() Element of bool REAL
- 1 is non empty V11() real ext-real non positive negative V85() Element of REAL
[.(- 1),1.] is V155() V156() V157() Element of bool REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
bool [:(Seg n),(Seg n):] is non empty V36() V40() set
p is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital V136() left_unital V189() V190() V191() doubleLoopStr
the carrier of p is non empty non trivial set
the carrier of p * is non empty functional FinSequence-membered FinSequenceSet of the carrier of p
q is Relation-like NAT -defined the carrier of p * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of p
Det q is Element of the carrier of p
Indices q is set
TR is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective V36() complex-yielding ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg n),(Seg n):]
q * TR is Relation-like NAT -defined Seg n -defined the carrier of p * -valued the carrier of p * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of p
(q * TR) @ is Relation-like NAT -defined the carrier of p * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of p
((q * TR) @) * TR is Relation-like NAT -defined Seg n -defined the carrier of p * -valued the carrier of p * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of p
(((q * TR) @) * TR) @ is Relation-like NAT -defined the carrier of p * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of p
Det ((((q * TR) @) * TR) @) is Element of the carrier of p
Permutations n is non empty permutational set
len (Permutations n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Seg (len (Permutations n)) is V16() V36() len (Permutations n) -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations n) ) } is set
- (Det q) is Element of the carrier of p
- (- (Det q)) is Element of the carrier of p
n1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective V36() complex-yielding ext-real-valued real-valued natural-valued finite-support Element of Permutations n
- ((Det q),n1) is Element of the carrier of p
Det (((q * TR) @) * TR) is Element of the carrier of p
Det ((q * TR) @) is Element of the carrier of p
- ((Det ((q * TR) @)),n1) is Element of the carrier of p
Det (q * TR) is Element of the carrier of p
- ((Det (q * TR)),n1) is Element of the carrier of p
- ((- ((Det q),n1)),n1) is Element of the carrier of p
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR . f is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
X is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[f,X] is set
{f,X} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{f} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{f,X},{f}} is non empty V36() V40() set
((((q * TR) @) * TR) @) * (f,X) is Element of the carrier of p
TR . X is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
q * ((TR . f),(TR . X)) is Element of the carrier of p
Indices ((((q * TR) @) * TR) @) is set
[X,f] is set
{X,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{X} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{X,f},{X}} is non empty V36() V40() set
Indices (((q * TR) @) * TR) is set
(((q * TR) @) * TR) * (X,f) is Element of the carrier of p
Indices ((q * TR) @) is set
[f,(TR . X)] is set
{f,(TR . X)} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{{f,(TR . X)},{f}} is non empty V36() V40() set
Indices (q * TR) is set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[z,f] is set
{z,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{z} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{z,f},{z}} is non empty V36() V40() set
((q * TR) @) * (z,f) is Element of the carrier of p
(q * TR) * (f,(TR . X)) is Element of the carrier of p
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[z,f] is set
{z,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{z} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{z,f},{z}} is non empty V36() V40() set
((q * TR) @) * (z,f) is Element of the carrier of p
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is non empty non degenerated non trivial right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital V136() left_unital V189() V190() V191() doubleLoopStr
the carrier of p is non empty non trivial set
the carrier of p * is non empty functional FinSequence-membered FinSequenceSet of the carrier of p
q is Relation-like NAT -defined the carrier of p * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular diagonal upper_triangular lower_triangular Matrix of n,n, the carrier of p
q @ is Relation-like NAT -defined the carrier of p * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of p
Indices q is set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[TR,n1] is set
{TR,n1} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{TR} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{TR,n1},{TR}} is non empty V36() V40() set
q * (TR,n1) is Element of the carrier of p
(q @) * (TR,n1) is Element of the carrier of p
[n1,TR] is set
{n1,TR} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{n1} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{n1,TR},{n1}} is non empty V36() V40() set
q * (n1,TR) is Element of the carrier of p
0. p is V61(p) Element of the carrier of p
the ZeroF of p is Element of the carrier of p
n is V11() real ext-real set
n ^2 is V11() real ext-real set
n * n is V11() real ext-real set
p is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
dom p is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
sqr p is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr p) is V11() real ext-real Element of REAL
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p +* (q,n) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
sqr (p +* (q,n)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (p +* (q,n))) is V11() real ext-real Element of REAL
p . q is V11() real ext-real Element of REAL
(p . q) ^2 is V11() real ext-real Element of REAL
(p . q) * (p . q) is V11() real ext-real set
(Sum (sqr p)) - ((p . q) ^2) is V11() real ext-real Element of REAL
- ((p . q) ^2) is V11() real ext-real set
(Sum (sqr p)) + (- ((p . q) ^2)) is V11() real ext-real set
((Sum (sqr p)) - ((p . q) ^2)) + (n ^2) is V11() real ext-real Element of REAL
@ p is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
@ (@ p) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
q -' 1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(@ (@ p)) | (q -' 1) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Seg (q -' 1) is V16() V36() q -' 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= q -' 1 ) } is set
(@ (@ p)) | (Seg (q -' 1)) is Relation-like NAT -defined Seg (q -' 1) -defined NAT -defined REAL -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
(@ (@ p)) /^ q is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
TR is V11() real ext-real Element of REAL
<*TR*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
[1,TR] is set
{1,TR} is non empty V36() V155() V156() V157() set
{{1,TR},{1}} is non empty V36() V40() set
{[1,TR]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
sqr <*TR*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
TR ^2 is V11() real ext-real Element of REAL
TR * TR is V11() real ext-real set
<*(TR ^2)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(TR ^2)] is set
{1,(TR ^2)} is non empty V36() V155() V156() V157() set
{{1,(TR ^2)},{1}} is non empty V36() V40() set
{[1,(TR ^2)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(@ (@ p)) +* (q,TR) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
((@ (@ p)) | (q -' 1)) ^ <*TR*> is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
(((@ (@ p)) | (q -' 1)) ^ <*TR*>) ^ ((@ (@ p)) /^ q) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
sqr (@ (@ p)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
sqr (((@ (@ p)) | (q -' 1)) ^ <*TR*>) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
sqr ((@ (@ p)) /^ q) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(sqr (((@ (@ p)) | (q -' 1)) ^ <*TR*>)) ^ (sqr ((@ (@ p)) /^ q)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
sqr ((@ (@ p)) | (q -' 1)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(sqr ((@ (@ p)) | (q -' 1))) ^ (sqr <*TR*>) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((sqr ((@ (@ p)) | (q -' 1))) ^ (sqr <*TR*>)) ^ (sqr ((@ (@ p)) /^ q)) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
Sum (sqr (@ (@ p))) is V11() real ext-real Element of REAL
Sum ((sqr ((@ (@ p)) | (q -' 1))) ^ (sqr <*TR*>)) is V11() real ext-real Element of REAL
Sum (sqr ((@ (@ p)) /^ q)) is V11() real ext-real Element of REAL
(Sum ((sqr ((@ (@ p)) | (q -' 1))) ^ (sqr <*TR*>))) + (Sum (sqr ((@ (@ p)) /^ q))) is V11() real ext-real Element of REAL
Sum (sqr ((@ (@ p)) | (q -' 1))) is V11() real ext-real Element of REAL
(Sum (sqr ((@ (@ p)) | (q -' 1)))) + (TR ^2) is V11() real ext-real Element of REAL
((Sum (sqr ((@ (@ p)) | (q -' 1)))) + (TR ^2)) + (Sum (sqr ((@ (@ p)) /^ q))) is V11() real ext-real Element of REAL
z is V11() real ext-real Element of REAL
<*z*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,z] is set
{1,z} is non empty V36() V155() V156() V157() set
{{1,z},{1}} is non empty V36() V40() set
{[1,z]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
sqr <*z*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
z ^2 is V11() real ext-real Element of REAL
z * z is V11() real ext-real set
<*(z ^2)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(z ^2)] is set
{1,(z ^2)} is non empty V36() V155() V156() V157() set
{{1,(z ^2)},{1}} is non empty V36() V40() set
{[1,(z ^2)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(@ (@ p)) +* (q,z) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
((@ (@ p)) | (q -' 1)) ^ <*z*> is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
(((@ (@ p)) | (q -' 1)) ^ <*z*>) ^ ((@ (@ p)) /^ q) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
sqr ((@ (@ p)) +* (q,z)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
sqr (((@ (@ p)) | (q -' 1)) ^ <*z*>) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(sqr (((@ (@ p)) | (q -' 1)) ^ <*z*>)) ^ (sqr ((@ (@ p)) /^ q)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
(sqr ((@ (@ p)) | (q -' 1))) ^ (sqr <*z*>) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((sqr ((@ (@ p)) | (q -' 1))) ^ (sqr <*z*>)) ^ (sqr ((@ (@ p)) /^ q)) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
Sum (sqr ((@ (@ p)) +* (q,z))) is V11() real ext-real Element of REAL
Sum ((sqr ((@ (@ p)) | (q -' 1))) ^ (sqr <*z*>)) is V11() real ext-real Element of REAL
(Sum ((sqr ((@ (@ p)) | (q -' 1))) ^ (sqr <*z*>))) + (Sum (sqr ((@ (@ p)) /^ q))) is V11() real ext-real Element of REAL
(Sum (sqr ((@ (@ p)) | (q -' 1)))) + (z ^2) is V11() real ext-real Element of REAL
((Sum (sqr ((@ (@ p)) | (q -' 1)))) + (z ^2)) + (Sum (sqr ((@ (@ p)) /^ q))) is V11() real ext-real Element of REAL
n is set
p is Relation-like Function-like Function-yielding V235() set
dom p is set
q is Relation-like Function-like set
p . q is Relation-like Function-like set
TR is set
(p . q) . TR is set
q . TR is set
n is set
bool n is non empty set
p is Element of bool n
q is Relation-like Function-like Function-yielding V235() set
TR is Relation-like Function-like set
dom q is set
q . TR is Relation-like Function-like set
n1 is set
(q . TR) . n1 is set
TR . n1 is set
n is set
p is set
n /\ p is set
q is Relation-like Function-like Function-yielding V235() set
TR is Relation-like Function-like set
dom q is set
q . TR is Relation-like Function-like set
n1 is set
(q . TR) . n1 is set
TR . n1 is set
n \/ p is set
TR is Relation-like Function-like Function-yielding V235() (p) set
q is Relation-like Function-like Function-yielding V235() (n) set
TR (#) q is Relation-like Function-like Function-yielding V235() set
f is Relation-like Function-like set
dom (TR (#) q) is set
(TR (#) q) . f is Relation-like Function-like set
X is set
((TR (#) q) . f) . X is set
f . X is set
dom TR is set
TR . f is Relation-like Function-like set
(TR . f) . X is set
q . (TR . f) is Relation-like Function-like set
dom q is set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
the non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total homogeneous Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total homogeneous Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL p):] is non empty set
q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL p):]
TR is set
dom q is non empty set
rng q is non empty set
q . TR is set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
q is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,p, the carrier of F_Real
Mx2Tran q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL p):]
the carrier of (TOP-REAL n) is non empty set
the carrier of (TOP-REAL p) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL p):] is non empty set
TR is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
n1 is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
TR + n1 is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) is non empty Relation-like [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total Element of bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):]
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
the addF of (TOP-REAL n) . (TR,n1) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
TR + n1 is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(Mx2Tran q) . (TR + n1) is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
(Mx2Tran q) . TR is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
(Mx2Tran q) . n1 is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
((Mx2Tran q) . TR) + ((Mx2Tran q) . n1) is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
the addF of (TOP-REAL p) is non empty Relation-like [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total Element of bool [:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):]
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):], the carrier of (TOP-REAL p):] is non empty set
the addF of (TOP-REAL p) . (((Mx2Tran q) . TR),((Mx2Tran q) . n1)) is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
((Mx2Tran q) . TR) + ((Mx2Tran q) . n1) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
q is V11() real ext-real set
TR is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
q * TR is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
q * TR is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(Mx2Tran p) . (q * TR) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(Mx2Tran p) . TR is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
q * ((Mx2Tran p) . TR) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
q * ((Mx2Tran p) . TR) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
p * q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
f is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
q . f is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
dom (p * q) is non empty set
n1 is V11() real ext-real set
n1 * f is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
n1 * f is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(p * q) . (n1 * f) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
q . (n1 * f) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p . (q . (n1 * f)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
X is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
n1 * X is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
n1 * X is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
p . (n1 * X) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p . X is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
n1 * (p . X) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
n1 * (p . X) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(p * q) . f is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
n1 * ((p * q) . f) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
n1 * ((p * q) . f) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
n1 is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
1. F_Real is V11() real ext-real V61( F_Real ) Element of the carrier of F_Real
the OneF of F_Real is V11() real ext-real Element of the carrier of F_Real
- (1. F_Real) is V11() real ext-real Element of the carrier of F_Real
K534((1. F_Real)) is V11() real ext-real Element of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Seg n1 is V16() V36() n1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n1 ) } is set
[:(Seg n1),(Seg n1):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
X is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[f,X] is set
{f,X} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{f} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{f,X},{f}} is non empty V36() V40() set
f is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n1,n1, the carrier of F_Real
Indices f is set
X is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
Indices X is set
[:(Seg p),(Seg p):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
[z,fp] is set
{z,fp} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{z} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{z,fp},{z}} is non empty V36() V40() set
X * (z,fp) is V11() real ext-real Element of the carrier of F_Real
diagonal_of_Matrix X is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
len (diagonal_of_Matrix X) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
fp + n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(diagonal_of_Matrix X) | (fp + n) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Seg (fp + n) is V16() V36() fp + n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= fp + n ) } is set
(diagonal_of_Matrix X) | (Seg (fp + n)) is Relation-like NAT -defined Seg (fp + n) -defined NAT -defined the carrier of F_Real -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
the multF of F_Real "**" ((diagonal_of_Matrix X) | (fp + n)) is V11() real ext-real Element of the carrier of F_Real
fp + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(fp + 1) + n is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() set
(diagonal_of_Matrix X) | ((fp + 1) + n) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Seg ((fp + 1) + n) is non empty V16() V36() (fp + 1) + n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= (fp + 1) + n ) } is set
(diagonal_of_Matrix X) | (Seg ((fp + 1) + n)) is Relation-like NAT -defined Seg ((fp + 1) + n) -defined NAT -defined the carrier of F_Real -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
the multF of F_Real "**" ((diagonal_of_Matrix X) | ((fp + 1) + n)) is V11() real ext-real Element of the carrier of F_Real
(fp + 1) + n is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(fp + n) + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (diagonal_of_Matrix X) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(diagonal_of_Matrix X) | ((fp + 1) + n) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Seg ((fp + 1) + n) is non empty V16() V36() (fp + 1) + n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= (fp + 1) + n ) } is set
(diagonal_of_Matrix X) | (Seg ((fp + 1) + n)) is Relation-like NAT -defined Seg ((fp + 1) + n) -defined NAT -defined the carrier of F_Real -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
(diagonal_of_Matrix X) . ((fp + 1) + n) is V11() real ext-real Element of REAL
<*((diagonal_of_Matrix X) . ((fp + 1) + n))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
[1,((diagonal_of_Matrix X) . ((fp + 1) + n))] is set
{1,((diagonal_of_Matrix X) . ((fp + 1) + n))} is non empty V36() V155() V156() V157() set
{{1,((diagonal_of_Matrix X) . ((fp + 1) + n))},{1}} is non empty V36() V40() set
{[1,((diagonal_of_Matrix X) . ((fp + 1) + n))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
((diagonal_of_Matrix X) | (fp + n)) ^ <*((diagonal_of_Matrix X) . ((fp + 1) + n))*> is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
[((fp + 1) + n),((fp + 1) + n)] is set
{((fp + 1) + n),((fp + 1) + n)} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{((fp + 1) + n)} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{((fp + 1) + n),((fp + 1) + n)},{((fp + 1) + n)}} is non empty V36() V40() set
X * (((fp + 1) + n),((fp + 1) + n)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real "**" ((diagonal_of_Matrix X) | ((fp + 1) + n)) is V11() real ext-real Element of the carrier of F_Real
(- (1. F_Real)) * (1. F_Real) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . ((- (1. F_Real)),(1. F_Real)) is V11() real ext-real Element of the carrier of F_Real
K538((- (1. F_Real)),(1. F_Real)) is V11() real ext-real Element of REAL
(diagonal_of_Matrix X) | {} is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued the carrier of F_Real -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() FinSequence of the carrier of F_Real
Seg {} is empty ordinal natural V11() real ext-real non positive non negative V16() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
(diagonal_of_Matrix X) | (Seg {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined Seg {} -defined NAT -defined RAT -valued the carrier of F_Real -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
the multF of F_Real "**" ((diagonal_of_Matrix X) | {}) is V11() real ext-real Element of the carrier of F_Real
<*> the carrier of F_Real is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined the carrier of F_Real -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of the carrier of F_Real *
the_unity_wrt the multF of F_Real is V11() real ext-real Element of the carrier of F_Real
p - n is V11() real ext-real V85() set
- n is V11() real ext-real non positive V85() set
p + (- n) is V11() real ext-real V85() set
(p - n) + n is V11() real ext-real V85() set
Det X is V11() real ext-real Element of the carrier of F_Real
X * (n,n) is V11() real ext-real Element of the carrier of F_Real
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(diagonal_of_Matrix X) | fp is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Seg fp is V16() V36() fp -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= fp ) } is set
(diagonal_of_Matrix X) | (Seg fp) is Relation-like NAT -defined Seg fp -defined NAT -defined the carrier of F_Real -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
the multF of F_Real "**" ((diagonal_of_Matrix X) | fp) is V11() real ext-real Element of the carrier of F_Real
fp + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(diagonal_of_Matrix X) | (fp + 1) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Seg (fp + 1) is non empty V16() V36() fp + 1 -element fp + 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= fp + 1 ) } is set
(diagonal_of_Matrix X) | (Seg (fp + 1)) is Relation-like NAT -defined Seg (fp + 1) -defined NAT -defined the carrier of F_Real -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
the multF of F_Real "**" ((diagonal_of_Matrix X) | (fp + 1)) is V11() real ext-real Element of the carrier of F_Real
dom (diagonal_of_Matrix X) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(diagonal_of_Matrix X) . (fp + 1) is V11() real ext-real Element of REAL
<*((diagonal_of_Matrix X) . (fp + 1))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
[1,((diagonal_of_Matrix X) . (fp + 1))] is set
{1,((diagonal_of_Matrix X) . (fp + 1))} is non empty V36() V155() V156() V157() set
{{1,((diagonal_of_Matrix X) . (fp + 1))},{1}} is non empty V36() V40() set
{[1,((diagonal_of_Matrix X) . (fp + 1))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
((diagonal_of_Matrix X) | fp) ^ <*((diagonal_of_Matrix X) . (fp + 1))*> is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
[(fp + 1),(fp + 1)] is set
{(fp + 1),(fp + 1)} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{(fp + 1)} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{(fp + 1),(fp + 1)},{(fp + 1)}} is non empty V36() V40() set
X * ((fp + 1),(fp + 1)) is V11() real ext-real Element of the carrier of F_Real
(1. F_Real) * (1. F_Real) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . ((1. F_Real),(1. F_Real)) is V11() real ext-real Element of the carrier of F_Real
K538((1. F_Real),(1. F_Real)) is V11() real ext-real Element of REAL
{} + n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(diagonal_of_Matrix X) | ({} + n) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Seg ({} + n) is V16() V36() {} + n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= {} + n ) } is set
(diagonal_of_Matrix X) | (Seg ({} + n)) is Relation-like NAT -defined Seg ({} + n) -defined NAT -defined the carrier of F_Real -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
the multF of F_Real "**" ((diagonal_of_Matrix X) | ({} + n)) is V11() real ext-real Element of the carrier of F_Real
n - 1 is V11() real ext-real V85() Element of REAL
n + (- 1) is V11() real ext-real V85() set
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
fp + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(diagonal_of_Matrix X) | fp is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Seg fp is V16() V36() fp -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= fp ) } is set
(diagonal_of_Matrix X) | (Seg fp) is Relation-like NAT -defined Seg fp -defined NAT -defined the carrier of F_Real -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
the multF of F_Real "**" ((diagonal_of_Matrix X) | fp) is V11() real ext-real Element of the carrier of F_Real
dom (diagonal_of_Matrix X) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(diagonal_of_Matrix X) | n is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(diagonal_of_Matrix X) | (Seg n) is Relation-like NAT -defined Seg n -defined NAT -defined the carrier of F_Real -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
(diagonal_of_Matrix X) . n is V11() real ext-real Element of REAL
<*((diagonal_of_Matrix X) . n)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
[1,((diagonal_of_Matrix X) . n)] is set
{1,((diagonal_of_Matrix X) . n)} is non empty V36() V155() V156() V157() set
{{1,((diagonal_of_Matrix X) . n)},{1}} is non empty V36() V40() set
{[1,((diagonal_of_Matrix X) . n)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
((diagonal_of_Matrix X) | fp) ^ <*((diagonal_of_Matrix X) . n)*> is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
[n,n] is set
{n,n} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{n} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{n,n},{n}} is non empty V36() V40() set
(1. F_Real) * (- (1. F_Real)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . ((1. F_Real),(- (1. F_Real))) is V11() real ext-real Element of the carrier of F_Real
K538((1. F_Real),(- (1. F_Real))) is V11() real ext-real Element of REAL
(diagonal_of_Matrix X) | p is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
(diagonal_of_Matrix X) | (Seg p) is Relation-like NAT -defined Seg p -defined NAT -defined the carrier of F_Real -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
the multF of F_Real "**" ((diagonal_of_Matrix X) | p) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real "**" (diagonal_of_Matrix X) is V11() real ext-real Element of the carrier of F_Real
[n,n] is set
{n,n} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{n} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{n,n},{n}} is non empty V36() V40() set
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
X * (fp,fp) is V11() real ext-real Element of the carrier of F_Real
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[fp,z] is set
{fp,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{fp} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{fp,z},{fp}} is non empty V36() V40() set
X * (fp,z) is V11() real ext-real Element of the carrier of F_Real
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
q is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Det q is V11() real ext-real Element of the carrier of F_Real
q * (p,p) is V11() real ext-real Element of the carrier of F_Real
Indices q is set
q is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
q * (p,p) is V11() real ext-real Element of the carrier of F_Real
Indices q is set
TR is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
TR * (p,p) is V11() real ext-real Element of the carrier of F_Real
Indices TR is set
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[n1,f] is set
{n1,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{n1} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{n1,f},{n1}} is non empty V36() V40() set
q * (n1,f) is V11() real ext-real Element of the carrier of F_Real
TR * (n1,f) is V11() real ext-real Element of the carrier of F_Real
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Det (p,n) is V11() real ext-real Element of the carrier of F_Real
q is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
Det q is V11() real ext-real Element of the carrier of F_Real
q * (n,n) is V11() real ext-real Element of the carrier of F_Real
Indices q is set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Col ((p,n),q) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (p,n) -element FinSequence-like FinSubsequence-like finite-support Element of (len (p,n)) -tuples_on the carrier of F_Real
len (p,n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len (p,n)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
TR is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
@ TR is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
(@ TR) "*" (Col ((p,n),q)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ TR),(Col ((p,n),q))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ TR),(Col ((p,n),q))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ TR),(Col ((p,n),q)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ TR),(Col ((p,n),q)))) is V11() real ext-real Element of the carrier of F_Real
TR . q is V11() real ext-real Element of REAL
Indices (p,n) is set
[:(Seg p),(Seg p):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
[q,q] is set
{q,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{q} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{q,q},{q}} is non empty V36() V40() set
dom (p,n) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
len (Col ((p,n),q)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (Col ((p,n),q)) is V36() len (p,n) -element V155() V156() V157() V158() V159() V160() Element of bool NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[z,q] is set
{z,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{z} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{z,q},{z}} is non empty V36() V40() set
(Col ((p,n),q)) . z is set
(p,n) * (z,q) is V11() real ext-real Element of the carrier of F_Real
len TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom TR is V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
(Col ((p,n),q)) . q is set
(p,n) * (q,q) is V11() real ext-real Element of the carrier of F_Real
mlt ((Col ((p,n),q)),(@ TR)) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(Col ((p,n),q)),(@ TR)) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((Col ((p,n),q)),(@ TR))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real "**" (mlt ((Col ((p,n),q)),(@ TR))) is V11() real ext-real Element of the carrier of F_Real
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Col ((p,n),n) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (p,n) -element FinSequence-like FinSubsequence-like finite-support Element of (len (p,n)) -tuples_on the carrier of F_Real
len (p,n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len (p,n)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
@ q is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
(@ q) "*" (Col ((p,n),n)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ q),(Col ((p,n),n))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ q),(Col ((p,n),n))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ q),(Col ((p,n),n)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ q),(Col ((p,n),n)))) is V11() real ext-real Element of the carrier of F_Real
q . n is V11() real ext-real Element of REAL
- (q . n) is V11() real ext-real Element of REAL
(@ q) . n is V11() real ext-real Element of REAL
dom (p,n) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(Col ((p,n),n)) . n is set
(p,n) * (n,n) is V11() real ext-real Element of the carrier of F_Real
len q is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (Col ((p,n),n)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (mlt ((@ q),(Col ((p,n),n)))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (mlt ((@ q),(Col ((p,n),n)))) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
Indices (p,n) is set
[:(Seg p),(Seg p):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(mlt ((@ q),(Col ((p,n),n)))) . z is V11() real ext-real Element of REAL
(@ q) . z is V11() real ext-real Element of REAL
[z,n] is set
{z,n} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{z} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{z,n},{z}} is non empty V36() V40() set
(Col ((p,n),n)) . z is set
(p,n) * (z,n) is V11() real ext-real Element of the carrier of F_Real
fp is V11() real ext-real Element of the carrier of F_Real
fp * ((p,n) * (z,n)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (fp,((p,n) * (z,n))) is V11() real ext-real Element of the carrier of F_Real
K538(fp,((p,n) * (z,n))) is V11() real ext-real Element of REAL
fp * (0. F_Real) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (fp,(0. F_Real)) is V11() real ext-real Element of the carrier of F_Real
K538(fp,(0. F_Real)) is V11() real ext-real Element of REAL
(mlt ((@ q),(Col ((p,n),n)))) . n is V11() real ext-real Element of REAL
n1 is V11() real ext-real Element of the carrier of F_Real
n1 * ((p,n) * (n,n)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (n1,((p,n) * (n,n))) is V11() real ext-real Element of the carrier of F_Real
K538(n1,((p,n) * (n,n))) is V11() real ext-real Element of REAL
n1 * (- (1. F_Real)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (n1,(- (1. F_Real))) is V11() real ext-real Element of the carrier of F_Real
K538(n1,(- (1. F_Real))) is V11() real ext-real Element of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
(p,n) ~ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
1. (F_Real,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
(p,n) * (p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
Indices (p,n) is set
X is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[X,z] is set
{X,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{X} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{X,z},{X}} is non empty V36() V40() set
(p,n) * (X,z) is V11() real ext-real Element of the carrier of F_Real
Indices ((p,n) * (p,n)) is set
X is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[X,z] is set
{X,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{X} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{X,z},{X}} is non empty V36() V40() set
((p,n) * (p,n)) * (X,z) is V11() real ext-real Element of the carrier of F_Real
(1. (F_Real,p)) * (X,z) is V11() real ext-real Element of the carrier of F_Real
width (p,n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Line ((p,n),X) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() width (p,n) -element FinSequence-like FinSubsequence-like finite-support Element of (width (p,n)) -tuples_on the carrier of F_Real
(width (p,n)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
@ (Line ((p,n),X)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
len (@ (Line ((p,n),X))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
len (p,n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
fp is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
@ fp is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Col ((p,n),z) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (p,n) -element FinSequence-like FinSubsequence-like finite-support Element of (len (p,n)) -tuples_on the carrier of F_Real
(len (p,n)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
(@ fp) "*" (Col ((p,n),z)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ fp),(Col ((p,n),z))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ fp),(Col ((p,n),z))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ fp),(Col ((p,n),z)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ fp),(Col ((p,n),z)))) is V11() real ext-real Element of the carrier of F_Real
[:(Seg p),(Seg p):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
(Line ((p,n),X)) . z is set
(p,n) * (X,z) is V11() real ext-real Element of the carrier of F_Real
Indices (1. (F_Real,p)) is set
- ((p,n) * (X,z)) is V11() real ext-real Element of the carrier of F_Real
K534(((p,n) * (X,z))) is V11() real ext-real Element of REAL
- (- (1. F_Real)) is V11() real ext-real Element of the carrier of F_Real
K534((- (1. F_Real))) is V11() real ext-real Element of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Mx2Tran (p,n) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
(Mx2Tran (p,n)) . TR is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
((Mx2Tran (p,n)) . TR) . q is V11() real ext-real Element of REAL
TR . q is V11() real ext-real Element of REAL
len ((Mx2Tran (p,n)) . TR) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom ((Mx2Tran (p,n)) . TR) is V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
len TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom TR is V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
@ TR is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Col ((p,n),q) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (p,n) -element FinSequence-like FinSubsequence-like finite-support Element of (len (p,n)) -tuples_on the carrier of F_Real
len (p,n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len (p,n)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
(@ TR) "*" (Col ((p,n),q)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ TR),(Col ((p,n),q))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ TR),(Col ((p,n),q))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ TR),(Col ((p,n),q)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ TR),(Col ((p,n),q)))) is V11() real ext-real Element of the carrier of F_Real
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Mx2Tran (p,n) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
(Mx2Tran (p,n)) . q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
((Mx2Tran (p,n)) . q) . n is V11() real ext-real Element of REAL
q . n is V11() real ext-real Element of REAL
- (q . n) is V11() real ext-real Element of REAL
@ q is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Col ((p,n),n) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (p,n) -element FinSequence-like FinSubsequence-like finite-support Element of (len (p,n)) -tuples_on the carrier of F_Real
len (p,n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len (p,n)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
(@ q) "*" (Col ((p,n),n)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ q),(Col ((p,n),n))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ q),(Col ((p,n),n))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ q),(Col ((p,n),n)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ q),(Col ((p,n),n)))) is V11() real ext-real Element of the carrier of F_Real
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Mx2Tran (p,n) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
(Mx2Tran (p,n)) . q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
q . n is V11() real ext-real Element of REAL
- (q . n) is V11() real ext-real Element of REAL
q +* (n,(- (q . n))) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
len q is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
X is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
((Mx2Tran (p,n)) . q) . X is V11() real ext-real Element of REAL
(q +* (n,(- (q . n)))) . X is V11() real ext-real Element of REAL
dom q is V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
q . X is V11() real ext-real Element of REAL
len (q +* (n,(- (q . n)))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len ((Mx2Tran (p,n)) . q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
{n} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Mx2Tran (p,n) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
TR is Relation-like Function-like set
dom (Mx2Tran (p,n)) is non empty set
(Mx2Tran (p,n)) . TR is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
n1 is set
((Mx2Tran (p,n)) . TR) . n1 is V11() real ext-real set
TR . n1 is set
((Mx2Tran (p,n)) . TR) . n1 is V11() real ext-real Element of REAL
f is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
(Mx2Tran (p,n)) . f is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
len ((Mx2Tran (p,n)) . f) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom ((Mx2Tran (p,n)) . f) is V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
len f is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom f is V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
((Mx2Tran (p,n)) . f) . n1 is V11() real ext-real Element of REAL
f . n1 is V11() real ext-real Element of REAL
n is V11() real ext-real set
cos n is V11() real ext-real set
sin n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
1. (F_Real,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
q is V11() real ext-real Element of the carrier of F_Real
TR is V11() real ext-real Element of the carrier of F_Real
- TR is V11() real ext-real Element of the carrier of F_Real
K534(TR) is V11() real ext-real Element of REAL
(q,TR) ][ ((- TR),q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of 2,2, the carrier of F_Real
<*((q,TR) ][ ((- TR),q)),(1. (F_Real,p))*> is non empty Relation-like NAT -defined ( the carrier of F_Real *) * -valued Function-like V36() 2 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of F_Real *) *
( the carrier of F_Real *) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real *
<*((q,TR) ][ ((- TR),q))*> is non empty trivial Relation-like NAT -defined Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() V282() set
[1,((q,TR) ][ ((- TR),q))] is set
{1,((q,TR) ][ ((- TR),q))} is non empty V36() V40() set
{{1,((q,TR) ][ ((- TR),q))},{1}} is non empty V36() V40() set
{[1,((q,TR) ][ ((- TR),q))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
<*(1. (F_Real,p))*> is non empty trivial Relation-like NAT -defined Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() V282() set
[1,(1. (F_Real,p))] is set
{1,(1. (F_Real,p))} is non empty V36() V40() set
{{1,(1. (F_Real,p))},{1}} is non empty V36() V40() set
{[1,(1. (F_Real,p))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
<*((q,TR) ][ ((- TR),q))*> ^ <*(1. (F_Real,p))*> is non empty Relation-like NAT -defined Function-like V36() 1 + 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() set
1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Len <*((q,TR) ][ ((- TR),q)),(1. (F_Real,p))*> is Relation-like NAT -defined NAT -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued natural-valued V198() finite-support Element of (len <*((q,TR) ][ ((- TR),q)),(1. (F_Real,p))*>) -tuples_on NAT
len <*((q,TR) ][ ((- TR),q)),(1. (F_Real,p))*> is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len <*((q,TR) ][ ((- TR),q)),(1. (F_Real,p))*>) -tuples_on NAT is functional FinSequence-membered FinSequenceSet of NAT
K910((Len <*((q,TR) ][ ((- TR),q)),(1. (F_Real,p))*>)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
block_diagonal (<*((q,TR) ][ ((- TR),q)),(1. (F_Real,p))*>,(0. F_Real)) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of K910((Len <*((q,TR) ][ ((- TR),q)),(1. (F_Real,p))*>)),K910((Len <*((q,TR) ][ ((- TR),q)),(1. (F_Real,p))*>)), the carrier of F_Real
Det (block_diagonal (<*((q,TR) ][ ((- TR),q)),(1. (F_Real,p))*>,(0. F_Real))) is V11() real ext-real Element of the carrier of F_Real
Det (1. (F_Real,p)) is V11() real ext-real Element of the carrier of F_Real
1_ F_Real is V11() real ext-real Element of the carrier of F_Real
(cos n) * (cos n) is V11() real ext-real set
(sin n) * (sin n) is V11() real ext-real set
((cos n) * (cos n)) + ((sin n) * (sin n)) is V11() real ext-real set
q * q is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
the multF of F_Real . (q,q) is V11() real ext-real Element of the carrier of F_Real
K538(q,q) is V11() real ext-real Element of REAL
TR * (- TR) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (TR,(- TR)) is V11() real ext-real Element of the carrier of F_Real
K538(TR,(- TR)) is V11() real ext-real Element of REAL
(q * q) - (TR * (- TR)) is V11() real ext-real Element of the carrier of F_Real
- (TR * (- TR)) is V11() real ext-real Element of the carrier of F_Real
K534((TR * (- TR))) is V11() real ext-real Element of REAL
(q * q) + (- (TR * (- TR))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real . ((q * q),(- (TR * (- TR)))) is V11() real ext-real Element of the carrier of F_Real
K536((q * q),(- (TR * (- TR)))) is V11() real ext-real Element of REAL
Det ((q,TR) ][ ((- TR),q)) is V11() real ext-real Element of the carrier of F_Real
cos . n is V11() real ext-real Element of REAL
sin . n is V11() real ext-real Element of REAL
(Det ((q,TR) ][ ((- TR),q))) * (Det (1. (F_Real,p))) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . ((Det ((q,TR) ][ ((- TR),q))),(Det (1. (F_Real,p)))) is V11() real ext-real Element of the carrier of F_Real
K538((Det ((q,TR) ][ ((- TR),q))),(Det (1. (F_Real,p)))) is V11() real ext-real Element of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Seg q is V16() V36() q -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= q ) } is set
[:(Seg q),(Seg q):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
bool [:(Seg q),(Seg q):] is non empty V36() V40() set
p + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
TR is non empty V155() V156() V157() V158() V159() V160() Element of bool NAT
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n1 - 2 is V11() real ext-real V85() Element of REAL
- 2 is non empty V11() real ext-real non positive negative V85() set
n1 + (- 2) is V11() real ext-real V85() set
n1 - 1 is V11() real ext-real V85() Element of REAL
n1 + (- 1) is V11() real ext-real V85() set
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
2 - 2 is V11() real ext-real V85() Element of REAL
2 + (- 2) is V11() real ext-real V85() set
n1 - {} is V11() real ext-real non negative V85() set
- {} is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
n1 + (- {}) is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
f + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(f + 1) + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n1 is Relation-like NAT -defined TR -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued natural-valued finite-support FinSequence of TR
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n1 /. 1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() Element of TR
1 - 2 is V11() real ext-real V85() Element of REAL
- 2 is non empty V11() real ext-real non positive negative V85() set
1 + (- 2) is V11() real ext-real V85() set
1 - 1 is V11() real ext-real V85() Element of REAL
1 + (- 1) is V11() real ext-real V85() set
1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n1 /. 2 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() Element of TR
2 - 2 is V11() real ext-real V85() Element of REAL
2 + (- 2) is V11() real ext-real V85() set
2 - 1 is V11() real ext-real V85() Element of REAL
2 + (- 1) is V11() real ext-real V85() set
dom n1 is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
n1 . 1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
rng n1 is V36() V155() V156() V157() V158() V159() V160() Element of bool REAL
[:TR,TR:] is non empty Relation-like RAT -valued INT -valued complex-yielding ext-real-valued real-valued natural-valued set
bool [:TR,TR:] is non empty set
n1 . 2 is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
X is set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
z + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(z + 1) + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z + 2 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
{} + 2 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n1 /. ((z + 1) + 1) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() Element of TR
((z + 1) + 1) - 2 is V11() real ext-real V85() Element of REAL
((z + 1) + 1) + (- 2) is V11() real ext-real V85() set
((z + 1) + 1) - 1 is V11() real ext-real V85() Element of REAL
((z + 1) + 1) + (- 1) is V11() real ext-real V85() set
n1 . ((z + 1) + 1) is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
n1 /. (z + 1) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() Element of TR
(z + 1) - 2 is V11() real ext-real V85() Element of REAL
(z + 1) + (- 2) is V11() real ext-real V85() set
(z + 1) - 1 is V11() real ext-real V85() Element of REAL
(z + 1) + (- 1) is V11() real ext-real V85() set
n1 . (z + 1) is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
n1 /. z is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() Element of TR
z - 2 is V11() real ext-real V85() Element of REAL
z + (- 2) is V11() real ext-real V85() set
z - 1 is V11() real ext-real V85() Element of REAL
z + (- 1) is V11() real ext-real V85() set
n1 . z is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
f is non empty Relation-like TR -defined TR -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued natural-valued Element of bool [:TR,TR:]
rng f is non empty V155() V156() V157() V158() V159() V160() Element of bool REAL
len TR is non empty ordinal cardinal set
X is Relation-like Seg q -defined Seg q -valued Function-like one-to-one total quasi_total onto bijective V36() complex-yielding ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg q),(Seg q):]
X . 1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
X . 2 is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
X . z is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
z - 2 is V11() real ext-real V85() Element of REAL
z + (- 2) is V11() real ext-real V85() set
z - 1 is V11() real ext-real V85() Element of REAL
z + (- 1) is V11() real ext-real V85() set
n1 /. z is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() Element of TR
n1 . z is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
n is V11() real ext-real set
cos n is V11() real ext-real set
sin n is V11() real ext-real set
- (sin n) is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Seg TR is V16() V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= TR ) } is set
TR - 2 is V11() real ext-real V85() Element of REAL
- 2 is non empty V11() real ext-real non positive negative V85() set
TR + (- 2) is V11() real ext-real V85() set
[:(Seg TR),(Seg TR):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
bool [:(Seg TR),(Seg TR):] is non empty V36() V40() set
p + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
fp is Relation-like Seg TR -defined Seg TR -valued Function-like one-to-one total quasi_total onto bijective V36() complex-yielding ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg TR),(Seg TR):]
fp . 1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
fp . 2 is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
dom fp is V36() set
z is Relation-like Function-like one-to-one set
z " is Relation-like Function-like one-to-one set
rng (z ") is set
rng fp is V36() V155() V156() V157() V158() V159() V160() Element of bool REAL
dom (z ") is set
fpz is Relation-like Seg TR -defined Seg TR -valued Function-like one-to-one total quasi_total V36() complex-yielding ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg TR),(Seg TR):]
h is Relation-like Seg TR -defined Seg TR -valued Function-like one-to-one total quasi_total onto bijective V36() complex-yielding ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg TR),(Seg TR):]
h . (fp . 1) is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
f is V11() real ext-real Element of the carrier of F_Real
n1 is V11() real ext-real Element of the carrier of F_Real
- n1 is V11() real ext-real Element of the carrier of F_Real
K534(n1) is V11() real ext-real Element of REAL
(f,n1) ][ ((- n1),f) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of 2,2, the carrier of F_Real
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
1. (F_Real,z) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of z,z, the carrier of F_Real
<*((f,n1) ][ ((- n1),f)),(1. (F_Real,z))*> is non empty Relation-like NAT -defined ( the carrier of F_Real *) * -valued Function-like V36() 2 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of F_Real *) *
( the carrier of F_Real *) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real *
<*((f,n1) ][ ((- n1),f))*> is non empty trivial Relation-like NAT -defined Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() V282() set
[1,((f,n1) ][ ((- n1),f))] is set
{1,((f,n1) ][ ((- n1),f))} is non empty V36() V40() set
{{1,((f,n1) ][ ((- n1),f))},{1}} is non empty V36() V40() set
{[1,((f,n1) ][ ((- n1),f))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
<*(1. (F_Real,z))*> is non empty trivial Relation-like NAT -defined Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() V282() set
[1,(1. (F_Real,z))] is set
{1,(1. (F_Real,z))} is non empty V36() V40() set
{{1,(1. (F_Real,z))},{1}} is non empty V36() V40() set
{[1,(1. (F_Real,z))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
<*((f,n1) ][ ((- n1),f))*> ^ <*(1. (F_Real,z))*> is non empty Relation-like NAT -defined Function-like V36() 1 + 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() set
1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
block_diagonal (<*((f,n1) ][ ((- n1),f)),(1. (F_Real,z))*>,(0. F_Real)) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of K910((Len <*((f,n1) ][ ((- n1),f)),(1. (F_Real,z))*>)),K910((Len <*((f,n1) ][ ((- n1),f)),(1. (F_Real,z))*>)), the carrier of F_Real
Len <*((f,n1) ][ ((- n1),f)),(1. (F_Real,z))*> is Relation-like NAT -defined NAT -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued natural-valued V198() finite-support Element of (len <*((f,n1) ][ ((- n1),f)),(1. (F_Real,z))*>) -tuples_on NAT
len <*((f,n1) ][ ((- n1),f)),(1. (F_Real,z))*> is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len <*((f,n1) ][ ((- n1),f)),(1. (F_Real,z))*>) -tuples_on NAT is functional FinSequence-membered FinSequenceSet of NAT
K910((Len <*((f,n1) ][ ((- n1),f)),(1. (F_Real,z))*>)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len ((f,n1) ][ ((- n1),f)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
<*((f,n1) ][ ((- n1),f))*> is non empty trivial Relation-like NAT -defined ( the carrier of F_Real *) * -valued Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() Matrix-yielding Square-Matrix-yielding V282() FinSequence of ( the carrier of F_Real *) *
Len <*((f,n1) ][ ((- n1),f))*> is Relation-like NAT -defined NAT -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued natural-valued V198() finite-support Element of (len <*((f,n1) ][ ((- n1),f))*>) -tuples_on NAT
len <*((f,n1) ][ ((- n1),f))*> is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len <*((f,n1) ][ ((- n1),f))*>) -tuples_on NAT is functional FinSequence-membered FinSequenceSet of NAT
<*2*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{[1,2]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
Sum (Len <*((f,n1) ][ ((- n1),f))*>) is V11() set
len (1. (F_Real,z)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Sum (Len <*((f,n1) ][ ((- n1),f)),(1. (F_Real,z))*>) is V11() set
2 + z is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
gf is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
Indices gf is set
h . (fp . 2) is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
gf * h is Relation-like NAT -defined Seg TR -defined the carrier of F_Real * -valued the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
(gf * h) @ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
((gf * h) @) * h is Relation-like NAT -defined Seg TR -defined the carrier of F_Real * -valued the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
(((gf * h) @) * h) @ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
dom h is V36() set
[p,p] is set
{p,p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{p} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{p,p},{p}} is non empty V36() V40() set
((((gf * h) @) * h) @) * (p,p) is V11() real ext-real Element of the carrier of F_Real
gf * (1,1) is V11() real ext-real Element of the carrier of F_Real
Det ((((gf * h) @) * h) @) is V11() real ext-real Element of the carrier of F_Real
((((gf * h) @) * h) @) * (q,q) is V11() real ext-real Element of the carrier of F_Real
((((gf * h) @) * h) @) * (p,q) is V11() real ext-real Element of the carrier of F_Real
((((gf * h) @) * h) @) * (q,p) is V11() real ext-real Element of the carrier of F_Real
Indices ((((gf * h) @) * h) @) is set
Det gf is V11() real ext-real Element of the carrier of F_Real
block_diagonal (<*((f,n1) ][ ((- n1),f))*>,(0. F_Real)) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of K910((Len <*((f,n1) ][ ((- n1),f))*>)),K910((Len <*((f,n1) ][ ((- n1),f))*>)), the carrier of F_Real
K910((Len <*((f,n1) ][ ((- n1),f))*>)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
width ((f,n1) ][ ((- n1),f)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Width <*((f,n1) ][ ((- n1),f))*> is Relation-like NAT -defined NAT -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued natural-valued V198() finite-support Element of (len <*((f,n1) ][ ((- n1),f))*>) -tuples_on NAT
Sum (Width <*((f,n1) ][ ((- n1),f))*>) is V11() set
[q,q] is set
{q,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{q} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{q,q},{q}} is non empty V36() V40() set
gf * (2,2) is V11() real ext-real Element of the carrier of F_Real
<*(1. (F_Real,z))*> is non empty trivial Relation-like NAT -defined ( the carrier of F_Real *) * -valued Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() Matrix-yielding Square-Matrix-yielding V282() FinSequence of ( the carrier of F_Real *) *
block_diagonal (<*(1. (F_Real,z))*>,(0. F_Real)) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of K910((Len <*(1. (F_Real,z))*>)),K910((Len <*(1. (F_Real,z))*>)), the carrier of F_Real
Len <*(1. (F_Real,z))*> is Relation-like NAT -defined NAT -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued natural-valued V198() finite-support Element of (len <*(1. (F_Real,z))*>) -tuples_on NAT
len <*(1. (F_Real,z))*> is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len <*(1. (F_Real,z))*>) -tuples_on NAT is functional FinSequence-membered FinSequenceSet of NAT
K910((Len <*(1. (F_Real,z))*>)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Indices ((f,n1) ][ ((- n1),f)) is set
((f,n1) ][ ((- n1),f)) * (1,1) is V11() real ext-real Element of the carrier of F_Real
gf * (2,1) is V11() real ext-real Element of the carrier of F_Real
((f,n1) ][ ((- n1),f)) * (2,1) is V11() real ext-real Element of the carrier of F_Real
gf * (1,2) is V11() real ext-real Element of the carrier of F_Real
((f,n1) ][ ((- n1),f)) * (1,2) is V11() real ext-real Element of the carrier of F_Real
((f,n1) ][ ((- n1),f)) * (2,2) is V11() real ext-real Element of the carrier of F_Real
<*((f,n1) ][ ((- n1),f))*> ^ <*(1. (F_Real,z))*> is non empty Relation-like NAT -defined ( the carrier of F_Real *) * -valued Function-like V36() 1 + 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() Matrix-yielding Square-Matrix-yielding FinSequence of ( the carrier of F_Real *) *
[p,q] is set
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{{p,q},{p}} is non empty V36() V40() set
[q,p] is set
{q,p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{{q,p},{q}} is non empty V36() V40() set
gfB is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
((((gf * h) @) * h) @) * (gfB,gfB) is V11() real ext-real Element of the carrier of F_Real
h is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[gfB,h] is set
{gfB,h} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{gfB} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{gfB,h},{gfB}} is non empty V36() V40() set
{gfB,h} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
((((gf * h) @) * h) @) * (gfB,h) is V11() real ext-real Element of the carrier of F_Real
h . gfB is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
h . h is ordinal natural V11() real ext-real non negative V36() cardinal V85() Element of REAL
rng h is V36() V155() V156() V157() V158() V159() V160() Element of bool REAL
[(h . gfB),(h . h)] is set
{(h . gfB),(h . h)} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{(h . gfB)} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{(h . gfB),(h . h)},{(h . gfB)}} is non empty V36() V40() set
gf * ((h . gfB),(h . h)) is V11() real ext-real Element of the carrier of F_Real
(h . gfB) - 2 is V11() real ext-real V85() Element of REAL
(h . gfB) + (- 2) is V11() real ext-real V85() set
i is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
i + 2 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
2 - 2 is V11() real ext-real V85() Element of REAL
2 + (- 2) is V11() real ext-real V85() set
[i,i] is set
{i,i} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{i} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{i,i},{i}} is non empty V36() V40() set
Indices (1. (F_Real,z)) is set
(1. (F_Real,z)) * (i,i) is V11() real ext-real Element of the carrier of F_Real
gf * ((i + 2),(i + 2)) is V11() real ext-real Element of the carrier of F_Real
(h . gfB) - 2 is V11() real ext-real V85() Element of REAL
(h . gfB) + (- 2) is V11() real ext-real V85() set
(h . h) - 2 is V11() real ext-real V85() Element of REAL
(h . h) + (- 2) is V11() real ext-real V85() set
i is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
2 - 2 is V11() real ext-real V85() Element of REAL
2 + (- 2) is V11() real ext-real V85() set
i + 2 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
H is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
H + 2 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
[i,H] is set
{i,H} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{i} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{i,H},{i}} is non empty V36() V40() set
Indices (1. (F_Real,z)) is set
(1. (F_Real,z)) * (i,H) is V11() real ext-real Element of the carrier of F_Real
gf * ((i + 2),(H + 2)) is V11() real ext-real Element of the carrier of F_Real
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is V11() real ext-real set
cos p is V11() real ext-real set
sin p is V11() real ext-real set
- (sin p) is V11() real ext-real set
{q,TR} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
n1 is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Det n1 is V11() real ext-real Element of the carrier of F_Real
n1 * (q,q) is V11() real ext-real Element of the carrier of F_Real
n1 * (TR,TR) is V11() real ext-real Element of the carrier of F_Real
n1 * (q,TR) is V11() real ext-real Element of the carrier of F_Real
n1 * (TR,q) is V11() real ext-real Element of the carrier of F_Real
Indices n1 is set
n1 is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
n1 * (q,q) is V11() real ext-real Element of the carrier of F_Real
n1 * (TR,TR) is V11() real ext-real Element of the carrier of F_Real
n1 * (q,TR) is V11() real ext-real Element of the carrier of F_Real
n1 * (TR,q) is V11() real ext-real Element of the carrier of F_Real
Indices n1 is set
f is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
f * (q,q) is V11() real ext-real Element of the carrier of F_Real
f * (TR,TR) is V11() real ext-real Element of the carrier of F_Real
f * (q,TR) is V11() real ext-real Element of the carrier of F_Real
f * (TR,q) is V11() real ext-real Element of the carrier of F_Real
Indices f is set
X is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[X,z] is set
{X,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{X} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{X,z},{X}} is non empty V36() V40() set
n1 * (X,z) is V11() real ext-real Element of the carrier of F_Real
f * (X,z) is V11() real ext-real Element of the carrier of F_Real
{X,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Det (TR,n,p,q) is V11() real ext-real Element of the carrier of F_Real
cos n is V11() real ext-real set
sin n is V11() real ext-real set
- (sin n) is V11() real ext-real set
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
n1 is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
Det n1 is V11() real ext-real Element of the carrier of F_Real
n1 * (p,p) is V11() real ext-real Element of the carrier of F_Real
n1 * (q,q) is V11() real ext-real Element of the carrier of F_Real
n1 * (p,q) is V11() real ext-real Element of the carrier of F_Real
n1 * (q,p) is V11() real ext-real Element of the carrier of F_Real
Indices n1 is set
n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
Seg TR is V16() V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= TR ) } is set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Col ((TR,n,p,q),n1) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (TR,n,p,q) -element FinSequence-like FinSubsequence-like finite-support Element of (len (TR,n,p,q)) -tuples_on the carrier of F_Real
len (TR,n,p,q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len (TR,n,p,q)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
f is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
@ f is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
(@ f) "*" (Col ((TR,n,p,q),n1)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ f),(Col ((TR,n,p,q),n1))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ f),(Col ((TR,n,p,q),n1))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ f),(Col ((TR,n,p,q),n1)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ f),(Col ((TR,n,p,q),n1)))) is V11() real ext-real Element of the carrier of F_Real
f . n1 is V11() real ext-real Element of REAL
Indices (TR,n,p,q) is set
[:(Seg TR),(Seg TR):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
[n1,n1] is set
{n1,n1} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{n1} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{n1,n1},{n1}} is non empty V36() V40() set
dom (TR,n,p,q) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
len (Col ((TR,n,p,q),n1)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (Col ((TR,n,p,q),n1)) is V36() len (TR,n,p,q) -element V155() V156() V157() V158() V159() V160() Element of bool NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[z,n1] is set
{z,n1} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{z} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{z,n1},{z}} is non empty V36() V40() set
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{z,n1} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
(Col ((TR,n,p,q),n1)) . z is set
(TR,n,p,q) * (z,n1) is V11() real ext-real Element of the carrier of F_Real
len f is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom f is V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
(Col ((TR,n,p,q),n1)) . n1 is set
(TR,n,p,q) * (n1,n1) is V11() real ext-real Element of the carrier of F_Real
mlt ((Col ((TR,n,p,q),n1)),(@ f)) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(Col ((TR,n,p,q),n1)),(@ f)) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((Col ((TR,n,p,q),n1)),(@ f))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real "**" (mlt ((Col ((TR,n,p,q),n1)),(@ f))) is V11() real ext-real Element of the carrier of F_Real
n is V11() real ext-real set
cos n is V11() real ext-real set
sin n is V11() real ext-real set
- (sin n) is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Col ((TR,n,p,q),p) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (TR,n,p,q) -element FinSequence-like FinSubsequence-like finite-support Element of (len (TR,n,p,q)) -tuples_on the carrier of F_Real
len (TR,n,p,q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len (TR,n,p,q)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
@ n1 is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
(@ n1) "*" (Col ((TR,n,p,q),p)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ n1),(Col ((TR,n,p,q),p))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ n1),(Col ((TR,n,p,q),p))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ n1),(Col ((TR,n,p,q),p)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ n1),(Col ((TR,n,p,q),p)))) is V11() real ext-real Element of the carrier of F_Real
n1 . p is V11() real ext-real Element of REAL
(n1 . p) * (cos n) is V11() real ext-real Element of REAL
n1 . q is V11() real ext-real Element of REAL
(n1 . q) * (- (sin n)) is V11() real ext-real Element of REAL
((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))) is V11() real ext-real Element of REAL
Seg TR is V16() V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= TR ) } is set
dom (TR,n,p,q) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(Col ((TR,n,p,q),p)) . q is set
(TR,n,p,q) * (q,p) is V11() real ext-real Element of the carrier of F_Real
(Col ((TR,n,p,q),p)) . p is set
(TR,n,p,q) * (p,p) is V11() real ext-real Element of the carrier of F_Real
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (Col ((TR,n,p,q),p)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (mlt ((@ n1),(Col ((TR,n,p,q),p)))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (mlt ((@ n1),(Col ((TR,n,p,q),p)))) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
Indices (TR,n,p,q) is set
[:(Seg TR),(Seg TR):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(mlt ((@ n1),(Col ((TR,n,p,q),p)))) . fp is V11() real ext-real Element of REAL
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fp,p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
(@ n1) . fp is V11() real ext-real Element of REAL
[fp,p] is set
{fp,p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{fp} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{fp,p},{fp}} is non empty V36() V40() set
(Col ((TR,n,p,q),p)) . fp is set
(TR,n,p,q) * (fp,p) is V11() real ext-real Element of the carrier of F_Real
z is V11() real ext-real Element of the carrier of F_Real
z * ((TR,n,p,q) * (fp,p)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (z,((TR,n,p,q) * (fp,p))) is V11() real ext-real Element of the carrier of F_Real
K538(z,((TR,n,p,q) * (fp,p))) is V11() real ext-real Element of REAL
z * (0. F_Real) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (z,(0. F_Real)) is V11() real ext-real Element of the carrier of F_Real
K538(z,(0. F_Real)) is V11() real ext-real Element of REAL
(mlt ((@ n1),(Col ((TR,n,p,q),p)))) /. p is V11() real ext-real Element of the carrier of F_Real
(mlt ((@ n1),(Col ((TR,n,p,q),p)))) /. q is V11() real ext-real Element of the carrier of F_Real
((mlt ((@ n1),(Col ((TR,n,p,q),p)))) /. p) + ((mlt ((@ n1),(Col ((TR,n,p,q),p)))) /. q) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real . (((mlt ((@ n1),(Col ((TR,n,p,q),p)))) /. p),((mlt ((@ n1),(Col ((TR,n,p,q),p)))) /. q)) is V11() real ext-real Element of the carrier of F_Real
K536(((mlt ((@ n1),(Col ((TR,n,p,q),p)))) /. p),((mlt ((@ n1),(Col ((TR,n,p,q),p)))) /. q)) is V11() real ext-real Element of REAL
(@ n1) . p is V11() real ext-real Element of REAL
(@ n1) . q is V11() real ext-real Element of REAL
(mlt ((@ n1),(Col ((TR,n,p,q),p)))) . p is V11() real ext-real Element of REAL
fp is V11() real ext-real Element of the carrier of F_Real
fp * ((TR,n,p,q) * (p,p)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (fp,((TR,n,p,q) * (p,p))) is V11() real ext-real Element of the carrier of F_Real
K538(fp,((TR,n,p,q) * (p,p))) is V11() real ext-real Element of REAL
(mlt ((@ n1),(Col ((TR,n,p,q),p)))) . q is V11() real ext-real Element of REAL
z is V11() real ext-real Element of the carrier of F_Real
z * ((TR,n,p,q) * (q,p)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (z,((TR,n,p,q) * (q,p))) is V11() real ext-real Element of the carrier of F_Real
K538(z,((TR,n,p,q) * (q,p))) is V11() real ext-real Element of REAL
n is V11() real ext-real set
sin n is V11() real ext-real set
cos n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Col ((TR,n,p,q),q) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (TR,n,p,q) -element FinSequence-like FinSubsequence-like finite-support Element of (len (TR,n,p,q)) -tuples_on the carrier of F_Real
len (TR,n,p,q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len (TR,n,p,q)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
@ n1 is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
(@ n1) "*" (Col ((TR,n,p,q),q)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ n1),(Col ((TR,n,p,q),q))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ n1),(Col ((TR,n,p,q),q))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ n1),(Col ((TR,n,p,q),q)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ n1),(Col ((TR,n,p,q),q)))) is V11() real ext-real Element of the carrier of F_Real
n1 . p is V11() real ext-real Element of REAL
(n1 . p) * (sin n) is V11() real ext-real Element of REAL
n1 . q is V11() real ext-real Element of REAL
(n1 . q) * (cos n) is V11() real ext-real Element of REAL
((n1 . p) * (sin n)) + ((n1 . q) * (cos n)) is V11() real ext-real Element of REAL
Seg TR is V16() V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= TR ) } is set
dom (TR,n,p,q) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(Col ((TR,n,p,q),q)) . q is set
(TR,n,p,q) * (q,q) is V11() real ext-real Element of the carrier of F_Real
(Col ((TR,n,p,q),q)) . p is set
(TR,n,p,q) * (p,q) is V11() real ext-real Element of the carrier of F_Real
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (Col ((TR,n,p,q),q)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (mlt ((@ n1),(Col ((TR,n,p,q),q)))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (mlt ((@ n1),(Col ((TR,n,p,q),q)))) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
Indices (TR,n,p,q) is set
[:(Seg TR),(Seg TR):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(mlt ((@ n1),(Col ((TR,n,p,q),q)))) . fp is V11() real ext-real Element of REAL
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fp,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
(@ n1) . fp is V11() real ext-real Element of REAL
[fp,q] is set
{fp,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{fp} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{fp,q},{fp}} is non empty V36() V40() set
(Col ((TR,n,p,q),q)) . fp is set
(TR,n,p,q) * (fp,q) is V11() real ext-real Element of the carrier of F_Real
z is V11() real ext-real Element of the carrier of F_Real
z * ((TR,n,p,q) * (fp,q)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (z,((TR,n,p,q) * (fp,q))) is V11() real ext-real Element of the carrier of F_Real
K538(z,((TR,n,p,q) * (fp,q))) is V11() real ext-real Element of REAL
z * (0. F_Real) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (z,(0. F_Real)) is V11() real ext-real Element of the carrier of F_Real
K538(z,(0. F_Real)) is V11() real ext-real Element of REAL
(mlt ((@ n1),(Col ((TR,n,p,q),q)))) /. p is V11() real ext-real Element of the carrier of F_Real
(mlt ((@ n1),(Col ((TR,n,p,q),q)))) /. q is V11() real ext-real Element of the carrier of F_Real
((mlt ((@ n1),(Col ((TR,n,p,q),q)))) /. p) + ((mlt ((@ n1),(Col ((TR,n,p,q),q)))) /. q) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real . (((mlt ((@ n1),(Col ((TR,n,p,q),q)))) /. p),((mlt ((@ n1),(Col ((TR,n,p,q),q)))) /. q)) is V11() real ext-real Element of the carrier of F_Real
K536(((mlt ((@ n1),(Col ((TR,n,p,q),q)))) /. p),((mlt ((@ n1),(Col ((TR,n,p,q),q)))) /. q)) is V11() real ext-real Element of REAL
(@ n1) . p is V11() real ext-real Element of REAL
(@ n1) . q is V11() real ext-real Element of REAL
(mlt ((@ n1),(Col ((TR,n,p,q),q)))) . p is V11() real ext-real Element of REAL
fp is V11() real ext-real Element of the carrier of F_Real
fp * ((TR,n,p,q) * (p,q)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (fp,((TR,n,p,q) * (p,q))) is V11() real ext-real Element of the carrier of F_Real
K538(fp,((TR,n,p,q) * (p,q))) is V11() real ext-real Element of REAL
(mlt ((@ n1),(Col ((TR,n,p,q),q)))) . q is V11() real ext-real Element of REAL
z is V11() real ext-real Element of the carrier of F_Real
z * ((TR,n,p,q) * (q,q)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real . (z,((TR,n,p,q) * (q,q))) is V11() real ext-real Element of the carrier of F_Real
K538(z,((TR,n,p,q) * (q,q))) is V11() real ext-real Element of REAL
n is V11() real ext-real set
p is V11() real ext-real set
n + p is V11() real ext-real set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(n1,n,q,TR) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n1,n1, the carrier of F_Real
(n1,p,q,TR) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n1,n1, the carrier of F_Real
(n1,n,q,TR) * (n1,p,q,TR) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n1,n1, the carrier of F_Real
(n1,(n + p),q,TR) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n1,n1, the carrier of F_Real
Seg n1 is V16() V36() n1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n1 ) } is set
width (n1,n,q,TR) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Indices (n1,n,q,TR) is set
[:(Seg n1),(Seg n1):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
Indices ((n1,n,q,TR) * (n1,p,q,TR)) is set
Indices (n1,(n + p),q,TR) is set
len (n1,p,q,TR) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
fpz is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
h is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[fpz,h] is set
{fpz,h} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{fpz} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{fpz,h},{fpz}} is non empty V36() V40() set
((n1,n,q,TR) * (n1,p,q,TR)) * (fpz,h) is V11() real ext-real Element of the carrier of F_Real
(n1,(n + p),q,TR) * (fpz,h) is V11() real ext-real Element of the carrier of F_Real
Line ((n1,n,q,TR),fpz) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() width (n1,n,q,TR) -element FinSequence-like FinSubsequence-like finite-support Element of (width (n1,n,q,TR)) -tuples_on the carrier of F_Real
(width (n1,n,q,TR)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
Col ((n1,p,q,TR),h) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (n1,p,q,TR) -element FinSequence-like FinSubsequence-like finite-support Element of (len (n1,p,q,TR)) -tuples_on the carrier of F_Real
(len (n1,p,q,TR)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
(Line ((n1,n,q,TR),fpz)) "*" (Col ((n1,p,q,TR),h)) is V11() real ext-real Element of the carrier of F_Real
mlt ((Line ((n1,n,q,TR),fpz)),(Col ((n1,p,q,TR),h))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(Line ((n1,n,q,TR),fpz)),(Col ((n1,p,q,TR),h))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((Line ((n1,n,q,TR),fpz)),(Col ((n1,p,q,TR),h)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((Line ((n1,n,q,TR),fpz)),(Col ((n1,p,q,TR),h)))) is V11() real ext-real Element of the carrier of F_Real
@ (Line ((n1,n,q,TR),fpz)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
len (@ (Line ((n1,n,q,TR),fpz))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
TOP-REAL n1 is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n1) is non empty set
sq is Relation-like NAT -defined Function-like V36() n1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n1)
sq . h is V11() real ext-real Element of REAL
(n1,n,q,TR) * (fpz,h) is V11() real ext-real Element of the carrier of F_Real
@ sq is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
sq . q is V11() real ext-real Element of REAL
(n1,n,q,TR) * (fpz,q) is V11() real ext-real Element of the carrier of F_Real
sq . TR is V11() real ext-real Element of REAL
(n1,n,q,TR) * (fpz,TR) is V11() real ext-real Element of the carrier of F_Real
cos p is V11() real ext-real set
(sq . q) * (cos p) is V11() real ext-real Element of REAL
sin p is V11() real ext-real set
- (sin p) is V11() real ext-real set
(sq . TR) * (- (sin p)) is V11() real ext-real Element of REAL
((sq . q) * (cos p)) + ((sq . TR) * (- (sin p))) is V11() real ext-real Element of REAL
cos n is V11() real ext-real set
(cos n) * (cos p) is V11() real ext-real set
((cos n) * (cos p)) + ((sq . TR) * (- (sin p))) is V11() real ext-real Element of REAL
sin n is V11() real ext-real set
(sin n) * (- (sin p)) is V11() real ext-real set
((cos n) * (cos p)) + ((sin n) * (- (sin p))) is V11() real ext-real set
(sin n) * (sin p) is V11() real ext-real set
((cos n) * (cos p)) - ((sin n) * (sin p)) is V11() real ext-real set
- ((sin n) * (sin p)) is V11() real ext-real set
((cos n) * (cos p)) + (- ((sin n) * (sin p))) is V11() real ext-real set
cos (n + p) is V11() real ext-real set
sin n is V11() real ext-real set
- (sin n) is V11() real ext-real set
(- (sin n)) * (cos p) is V11() real ext-real set
((- (sin n)) * (cos p)) + ((sq . TR) * (- (sin p))) is V11() real ext-real Element of REAL
cos n is V11() real ext-real set
(cos n) * (- (sin p)) is V11() real ext-real set
((- (sin n)) * (cos p)) + ((cos n) * (- (sin p))) is V11() real ext-real set
(sin n) * (cos p) is V11() real ext-real set
(cos n) * (sin p) is V11() real ext-real set
((sin n) * (cos p)) + ((cos n) * (sin p)) is V11() real ext-real set
- (((sin n) * (cos p)) + ((cos n) * (sin p))) is V11() real ext-real set
sin (n + p) is V11() real ext-real set
- (sin (n + p)) is V11() real ext-real set
{q,TR} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fpz,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fpz,TR} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
[fpz,TR] is set
{fpz,TR} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{{fpz,TR},{fpz}} is non empty V36() V40() set
[fpz,q] is set
{fpz,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{{fpz,q},{fpz}} is non empty V36() V40() set
{} * (cos p) is V11() real ext-real set
{} * (- (sin p)) is V11() real ext-real set
({} * (cos p)) + ({} * (- (sin p))) is V11() real ext-real set
sin p is V11() real ext-real set
(sq . q) * (sin p) is V11() real ext-real Element of REAL
cos p is V11() real ext-real set
(sq . TR) * (cos p) is V11() real ext-real Element of REAL
((sq . q) * (sin p)) + ((sq . TR) * (cos p)) is V11() real ext-real Element of REAL
cos n is V11() real ext-real set
(cos n) * (sin p) is V11() real ext-real set
((cos n) * (sin p)) + ((sq . TR) * (cos p)) is V11() real ext-real Element of REAL
sin n is V11() real ext-real set
(sin n) * (cos p) is V11() real ext-real set
((cos n) * (sin p)) + ((sin n) * (cos p)) is V11() real ext-real set
sin (n + p) is V11() real ext-real set
sin n is V11() real ext-real set
- (sin n) is V11() real ext-real set
(- (sin n)) * (sin p) is V11() real ext-real set
((- (sin n)) * (sin p)) + ((sq . TR) * (cos p)) is V11() real ext-real Element of REAL
cos n is V11() real ext-real set
(cos n) * (cos p) is V11() real ext-real set
(sin n) * (sin p) is V11() real ext-real set
((cos n) * (cos p)) - ((sin n) * (sin p)) is V11() real ext-real set
- ((sin n) * (sin p)) is V11() real ext-real set
((cos n) * (cos p)) + (- ((sin n) * (sin p))) is V11() real ext-real set
cos (n + p) is V11() real ext-real set
{q,TR} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fpz,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fpz,TR} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
[fpz,TR] is set
{fpz,TR} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{{fpz,TR},{fpz}} is non empty V36() V40() set
[fpz,q] is set
{fpz,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{{fpz,q},{fpz}} is non empty V36() V40() set
{} * (sin p) is V11() real ext-real set
{} * (cos p) is V11() real ext-real set
({} * (sin p)) + ({} * (cos p)) is V11() real ext-real set
{q,TR} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fpz,h} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
n is V11() real ext-real set
- n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
(TR,n,p,q) @ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
(TR,(- n),p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Seg TR is V16() V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= TR ) } is set
Indices (TR,(- n),p,q) is set
[:(Seg TR),(Seg TR):] is Relation-like RAT -valued INT -valued V36() complex-yielding ext-real-valued real-valued natural-valued set
Indices (TR,n,p,q) is set
Indices ((TR,n,p,q) @) is set
cos n is V11() real ext-real set
cos (- n) is V11() real ext-real set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[z,fp] is set
{z,fp} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{z} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{z,fp},{z}} is non empty V36() V40() set
((TR,n,p,q) @) * (z,fp) is V11() real ext-real Element of the carrier of F_Real
(TR,(- n),p,q) * (z,fp) is V11() real ext-real Element of the carrier of F_Real
sin n is V11() real ext-real set
- (sin n) is V11() real ext-real set
sin (- n) is V11() real ext-real set
[fp,z] is set
{fp,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{fp} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{fp,z},{fp}} is non empty V36() V40() set
(TR,n,p,q) * (fp,z) is V11() real ext-real Element of the carrier of F_Real
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fp,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
- (sin (- n)) is V11() real ext-real set
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fp,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fp,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{fp,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(q,{},n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of q,q, the carrier of F_Real
1. (F_Real,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of q,q, the carrier of F_Real
Indices (q,{},n,p) is set
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[n1,f] is set
{n1,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{n1} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{n1,f},{n1}} is non empty V36() V40() set
(q,{},n,p) * (n1,f) is V11() real ext-real Element of the carrier of F_Real
sin {} is V11() real ext-real set
- (sin {}) is V11() real ext-real set
{n,p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{n1,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{n,p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{n1,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{n,p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{n1,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{n,p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{n1,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{n,p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{n1,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
[n1,n1] is set
{n1,n1} is non empty V36() V40() V155() V156() V157() V158() V159() V160() set
{n1} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() set
{{n1,n1},{n1}} is non empty V36() V40() set
(q,{},n,p) * (n1,n1) is V11() real ext-real Element of the carrier of F_Real
n is V11() real ext-real set
- n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
(TR,n,p,q) ~ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
(TR,(- n),p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
(TR,n,p,q) * (TR,(- n),p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
n + (- n) is V11() real ext-real set
(TR,(n + (- n)),p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
1. (F_Real,TR) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
(TR,(- n),p,q) * (TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
(- n) + n is V11() real ext-real set
(TR,((- n) + n),p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
n is V11() real ext-real set
- n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
(TR,n,p,q) ~ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
(TR,(- n),p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
(TR,n,p,q) @ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,n,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
f is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
(Mx2Tran (TR,n,p,q)) . f is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
((Mx2Tran (TR,n,p,q)) . f) . n1 is V11() real ext-real Element of REAL
f . n1 is V11() real ext-real Element of REAL
Seg TR is V16() V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= TR ) } is set
len ((Mx2Tran (TR,n,p,q)) . f) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom ((Mx2Tran (TR,n,p,q)) . f) is V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
len f is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom f is V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
@ f is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Col ((TR,n,p,q),n1) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (TR,n,p,q) -element FinSequence-like FinSubsequence-like finite-support Element of (len (TR,n,p,q)) -tuples_on the carrier of F_Real
len (TR,n,p,q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len (TR,n,p,q)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
(@ f) "*" (Col ((TR,n,p,q),n1)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ f),(Col ((TR,n,p,q),n1))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ f),(Col ((TR,n,p,q),n1))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ f),(Col ((TR,n,p,q),n1)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ f),(Col ((TR,n,p,q),n1)))) is V11() real ext-real Element of the carrier of F_Real
n is V11() real ext-real set
cos n is V11() real ext-real set
sin n is V11() real ext-real set
- (sin n) is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,n,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
(Mx2Tran (TR,n,p,q)) . n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
((Mx2Tran (TR,n,p,q)) . n1) . p is V11() real ext-real Element of REAL
n1 . p is V11() real ext-real Element of REAL
(n1 . p) * (cos n) is V11() real ext-real Element of REAL
n1 . q is V11() real ext-real Element of REAL
(n1 . q) * (- (sin n)) is V11() real ext-real Element of REAL
((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))) is V11() real ext-real Element of REAL
Seg TR is V16() V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= TR ) } is set
@ n1 is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Col ((TR,n,p,q),p) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (TR,n,p,q) -element FinSequence-like FinSubsequence-like finite-support Element of (len (TR,n,p,q)) -tuples_on the carrier of F_Real
len (TR,n,p,q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len (TR,n,p,q)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
(@ n1) "*" (Col ((TR,n,p,q),p)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ n1),(Col ((TR,n,p,q),p))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ n1),(Col ((TR,n,p,q),p))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ n1),(Col ((TR,n,p,q),p)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ n1),(Col ((TR,n,p,q),p)))) is V11() real ext-real Element of the carrier of F_Real
n is V11() real ext-real set
sin n is V11() real ext-real set
cos n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,n,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
(Mx2Tran (TR,n,p,q)) . n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
((Mx2Tran (TR,n,p,q)) . n1) . q is V11() real ext-real Element of REAL
n1 . p is V11() real ext-real Element of REAL
(n1 . p) * (sin n) is V11() real ext-real Element of REAL
n1 . q is V11() real ext-real Element of REAL
(n1 . q) * (cos n) is V11() real ext-real Element of REAL
((n1 . p) * (sin n)) + ((n1 . q) * (cos n)) is V11() real ext-real Element of REAL
Seg TR is V16() V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= TR ) } is set
@ n1 is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Col ((TR,n,p,q),q) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() len (TR,n,p,q) -element FinSequence-like FinSubsequence-like finite-support Element of (len (TR,n,p,q)) -tuples_on the carrier of F_Real
len (TR,n,p,q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(len (TR,n,p,q)) -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
(@ n1) "*" (Col ((TR,n,p,q),q)) is V11() real ext-real Element of the carrier of F_Real
mlt ((@ n1),(Col ((TR,n,p,q),q))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the multF of F_Real,(@ n1),(Col ((TR,n,p,q),q))) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
Sum (mlt ((@ n1),(Col ((TR,n,p,q),q)))) is V11() real ext-real Element of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
the addF of F_Real "**" (mlt ((@ n1),(Col ((TR,n,p,q),q)))) is V11() real ext-real Element of the carrier of F_Real
n is V11() real ext-real set
cos n is V11() real ext-real set
sin n is V11() real ext-real set
- (sin n) is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p -' 1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q -' p is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(q -' p) -' 1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,n,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
(Mx2Tran (TR,n,p,q)) . n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
n1 | (p -' 1) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
Seg (p -' 1) is V16() V36() p -' 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p -' 1 ) } is set
n1 | (Seg (p -' 1)) is Relation-like NAT -defined Seg (p -' 1) -defined NAT -defined Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
n1 . p is V11() real ext-real Element of REAL
(n1 . p) * (cos n) is V11() real ext-real Element of REAL
n1 . q is V11() real ext-real Element of REAL
(n1 . q) * (- (sin n)) is V11() real ext-real Element of REAL
((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))) is V11() real ext-real Element of REAL
<*(((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
[1,(((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))))] is set
{1,(((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))))} is non empty V36() V155() V156() V157() set
{{1,(((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))))},{1}} is non empty V36() V40() set
{[1,(((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(n1 | (p -' 1)) ^ <*(((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))))*> is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
n1 /^ p is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
(n1 /^ p) | ((q -' p) -' 1) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
Seg ((q -' p) -' 1) is V16() V36() (q -' p) -' 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= (q -' p) -' 1 ) } is set
(n1 /^ p) | (Seg ((q -' p) -' 1)) is Relation-like NAT -defined Seg ((q -' p) -' 1) -defined NAT -defined Function-like V36() FinSubsequence-like finite-support set
((n1 | (p -' 1)) ^ <*(((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))))*>) ^ ((n1 /^ p) | ((q -' p) -' 1)) is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
(n1 . p) * (sin n) is V11() real ext-real Element of REAL
(n1 . q) * (cos n) is V11() real ext-real Element of REAL
((n1 . p) * (sin n)) + ((n1 . q) * (cos n)) is V11() real ext-real Element of REAL
<*(((n1 . p) * (sin n)) + ((n1 . q) * (cos n)))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(((n1 . p) * (sin n)) + ((n1 . q) * (cos n)))] is set
{1,(((n1 . p) * (sin n)) + ((n1 . q) * (cos n)))} is non empty V36() V155() V156() V157() set
{{1,(((n1 . p) * (sin n)) + ((n1 . q) * (cos n)))},{1}} is non empty V36() V40() set
{[1,(((n1 . p) * (sin n)) + ((n1 . q) * (cos n)))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(((n1 | (p -' 1)) ^ <*(((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))))*>) ^ ((n1 /^ p) | ((q -' p) -' 1))) ^ <*(((n1 . p) * (sin n)) + ((n1 . q) * (cos n)))*> is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
n1 /^ q is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
((((n1 | (p -' 1)) ^ <*(((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))))*>) ^ ((n1 /^ p) | ((q -' p) -' 1))) ^ <*(((n1 . p) * (sin n)) + ((n1 . q) * (cos n)))*>) ^ (n1 /^ q) is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
(p -' 1) + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
p - 1 is V11() real ext-real V85() Element of REAL
p + (- 1) is V11() real ext-real V85() set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
((Mx2Tran (TR,n,p,q)) . n1) | (p -' 1) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
((Mx2Tran (TR,n,p,q)) . n1) | (Seg (p -' 1)) is Relation-like NAT -defined Seg (p -' 1) -defined NAT -defined Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
(((Mx2Tran (TR,n,p,q)) . n1) | (p -' 1)) . z is set
((Mx2Tran (TR,n,p,q)) . n1) . z is V11() real ext-real Element of REAL
(n1 | (p -' 1)) . z is set
n1 . z is V11() real ext-real Element of REAL
len ((Mx2Tran (TR,n,p,q)) . n1) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (((Mx2Tran (TR,n,p,q)) . n1) | (p -' 1)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (n1 /^ p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
TR - p is V11() real ext-real V85() set
- p is V11() real ext-real non positive V85() set
TR + (- p) is V11() real ext-real V85() set
q - p is V11() real ext-real V85() set
q + (- p) is V11() real ext-real V85() set
((q -' p) -' 1) + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
p - p is V11() real ext-real V85() set
p + (- p) is V11() real ext-real V85() set
(q -' p) - 1 is V11() real ext-real V85() Element of REAL
(q -' p) + (- 1) is V11() real ext-real V85() set
len (n1 /^ q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
TR - q is V11() real ext-real V85() set
- q is V11() real ext-real non positive V85() set
TR + (- q) is V11() real ext-real V85() set
((Mx2Tran (TR,n,p,q)) . n1) /^ p is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
len (((Mx2Tran (TR,n,p,q)) . n1) /^ p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(((Mx2Tran (TR,n,p,q)) . n1) /^ p) | ((q -' p) -' 1) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
(((Mx2Tran (TR,n,p,q)) . n1) /^ p) | (Seg ((q -' p) -' 1)) is Relation-like NAT -defined Seg ((q -' p) -' 1) -defined NAT -defined Function-like V36() FinSubsequence-like finite-support set
len ((((Mx2Tran (TR,n,p,q)) . n1) /^ p) | ((q -' p) -' 1)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
((n1 /^ p) | ((q -' p) -' 1)) . z is set
(n1 /^ p) . z is set
dom (((Mx2Tran (TR,n,p,q)) . n1) /^ p) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(((Mx2Tran (TR,n,p,q)) . n1) /^ p) . z is set
p + z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
((Mx2Tran (TR,n,p,q)) . n1) . (p + z) is V11() real ext-real Element of REAL
p + (q -' p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (n1 /^ p) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
n1 . (p + z) is V11() real ext-real Element of REAL
z + p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
((((Mx2Tran (TR,n,p,q)) . n1) /^ p) | ((q -' p) -' 1)) . z is set
len ((n1 /^ p) | ((q -' p) -' 1)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
((Mx2Tran (TR,n,p,q)) . n1) /^ q is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
len (((Mx2Tran (TR,n,p,q)) . n1) /^ q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
dom (((Mx2Tran (TR,n,p,q)) . n1) /^ q) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(((Mx2Tran (TR,n,p,q)) . n1) /^ q) . z is set
q + z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
((Mx2Tran (TR,n,p,q)) . n1) . (q + z) is V11() real ext-real Element of REAL
dom (n1 /^ q) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(n1 /^ q) . z is set
n1 . (q + z) is V11() real ext-real Element of REAL
len (n1 | (p -' 1)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
((Mx2Tran (TR,n,p,q)) . n1) . p is V11() real ext-real Element of REAL
@ ((Mx2Tran (TR,n,p,q)) . n1) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
@ (@ ((Mx2Tran (TR,n,p,q)) . n1)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
<*(((Mx2Tran (TR,n,p,q)) . n1) . p)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(((Mx2Tran (TR,n,p,q)) . n1) . p)] is set
{1,(((Mx2Tran (TR,n,p,q)) . n1) . p)} is non empty V36() V155() V156() V157() set
{{1,(((Mx2Tran (TR,n,p,q)) . n1) . p)},{1}} is non empty V36() V40() set
{[1,(((Mx2Tran (TR,n,p,q)) . n1) . p)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(((Mx2Tran (TR,n,p,q)) . n1) | (p -' 1)) ^ <*(((Mx2Tran (TR,n,p,q)) . n1) . p)*> is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
((((Mx2Tran (TR,n,p,q)) . n1) | (p -' 1)) ^ <*(((Mx2Tran (TR,n,p,q)) . n1) . p)*>) ^ ((((Mx2Tran (TR,n,p,q)) . n1) /^ p) | ((q -' p) -' 1)) is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
((Mx2Tran (TR,n,p,q)) . n1) . q is V11() real ext-real Element of REAL
<*(((Mx2Tran (TR,n,p,q)) . n1) . q)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(((Mx2Tran (TR,n,p,q)) . n1) . q)] is set
{1,(((Mx2Tran (TR,n,p,q)) . n1) . q)} is non empty V36() V155() V156() V157() set
{{1,(((Mx2Tran (TR,n,p,q)) . n1) . q)},{1}} is non empty V36() V40() set
{[1,(((Mx2Tran (TR,n,p,q)) . n1) . q)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(((((Mx2Tran (TR,n,p,q)) . n1) | (p -' 1)) ^ <*(((Mx2Tran (TR,n,p,q)) . n1) . p)*>) ^ ((((Mx2Tran (TR,n,p,q)) . n1) /^ p) | ((q -' p) -' 1))) ^ <*(((Mx2Tran (TR,n,p,q)) . n1) . q)*> is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
((((((Mx2Tran (TR,n,p,q)) . n1) | (p -' 1)) ^ <*(((Mx2Tran (TR,n,p,q)) . n1) . p)*>) ^ ((((Mx2Tran (TR,n,p,q)) . n1) /^ p) | ((q -' p) -' 1))) ^ <*(((Mx2Tran (TR,n,p,q)) . n1) . q)*>) ^ (((Mx2Tran (TR,n,p,q)) . n1) /^ q) is non empty Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like finite-support set
n is V11() real ext-real set
n ^2 is V11() real ext-real set
n * n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
n1 . p is V11() real ext-real Element of REAL
(n1 . p) ^2 is V11() real ext-real Element of REAL
(n1 . p) * (n1 . p) is V11() real ext-real set
n1 . q is V11() real ext-real Element of REAL
(n1 . q) ^2 is V11() real ext-real Element of REAL
(n1 . q) * (n1 . q) is V11() real ext-real set
((n1 . p) ^2) + ((n1 . q) ^2) is V11() real ext-real Element of REAL
sqrt (((n1 . p) ^2) + ((n1 . q) ^2)) is V11() real ext-real Element of REAL
(n1 . p) * (n1 . p) is V11() real ext-real Element of REAL
(n1 . q) * (n1 . q) is V11() real ext-real Element of REAL
n / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
(sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) " is V11() real ext-real set
n * ((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) ") is V11() real ext-real set
(n / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
((n1 . p) ^2) + {} is V11() real ext-real Element of REAL
sqrt ((n1 . p) ^2) is V11() real ext-real Element of REAL
(n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
(n1 . p) * ((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) ") is V11() real ext-real set
- (n1 . p) is V11() real ext-real Element of REAL
(n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
(n1 . p) * ((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) ") is V11() real ext-real set
(n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
(n1 . p) * ((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) ") is V11() real ext-real set
(n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
(n1 . p) * ((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) ") is V11() real ext-real set
dom sin is non empty set
z is set
sin . z is V11() real ext-real Element of REAL
(sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
(sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) ^2 is V11() real ext-real Element of REAL
(sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real set
sqrt (n ^2) is V11() real ext-real set
- n is V11() real ext-real set
fpz is set
sin . fpz is V11() real ext-real Element of REAL
(sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . p) is V11() real ext-real Element of REAL
((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . p)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . p)) * ((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) ") is V11() real ext-real set
sq is V11() real ext-real Element of REAL
sin . sq is V11() real ext-real Element of REAL
sin sq is V11() real ext-real Element of REAL
((n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * ((n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) is V11() real ext-real Element of REAL
((n1 . p) * (n1 . p)) / (((n1 . p) ^2) + ((n1 . q) ^2)) is V11() real ext-real Element of REAL
(((n1 . p) ^2) + ((n1 . q) ^2)) " is V11() real ext-real set
((n1 . p) * (n1 . p)) * ((((n1 . p) ^2) + ((n1 . q) ^2)) ") is V11() real ext-real set
cos sq is V11() real ext-real Element of REAL
(cos sq) * (cos sq) is V11() real ext-real Element of REAL
((cos sq) * (cos sq)) + (((n1 . p) * (n1 . p)) / (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
1 - (((n1 . p) * (n1 . p)) / (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
- (((n1 . p) * (n1 . p)) / (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real set
1 + (- (((n1 . p) * (n1 . p)) / (((n1 . p) ^2) + ((n1 . q) ^2)))) is V11() real ext-real set
(((n1 . p) ^2) + ((n1 . q) ^2)) / (((n1 . p) ^2) + ((n1 . q) ^2)) is V11() real ext-real Element of REAL
(((n1 . p) ^2) + ((n1 . q) ^2)) * ((((n1 . p) ^2) + ((n1 . q) ^2)) ") is V11() real ext-real set
((((n1 . p) ^2) + ((n1 . q) ^2)) / (((n1 . p) ^2) + ((n1 . q) ^2))) - (((n1 . p) * (n1 . p)) / (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
((((n1 . p) ^2) + ((n1 . q) ^2)) / (((n1 . p) ^2) + ((n1 . q) ^2))) + (- (((n1 . p) * (n1 . p)) / (((n1 . p) ^2) + ((n1 . q) ^2)))) is V11() real ext-real set
((n1 . q) * (n1 . q)) / (((n1 . p) ^2) + ((n1 . q) ^2)) is V11() real ext-real Element of REAL
((n1 . q) * (n1 . q)) * ((((n1 . p) ^2) + ((n1 . q) ^2)) ") is V11() real ext-real set
(n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
(n1 . q) * ((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) ") is V11() real ext-real set
((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) ^2 is V11() real ext-real Element of REAL
((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * ((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) is V11() real ext-real set
(cos sq) ^2 is V11() real ext-real Element of REAL
(cos sq) * (cos sq) is V11() real ext-real set
h is V11() real ext-real Element of REAL
sq - h is V11() real ext-real Element of REAL
- h is V11() real ext-real set
sq + (- h) is V11() real ext-real set
z is V11() real ext-real Element of REAL
(TR,z,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,z,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
(Mx2Tran (TR,z,p,q)) . n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
((Mx2Tran (TR,z,p,q)) . n1) . p is V11() real ext-real Element of REAL
sin h is V11() real ext-real Element of REAL
- (sin h) is V11() real ext-real Element of REAL
- sq is V11() real ext-real Element of REAL
(- sq) + z is V11() real ext-real Element of REAL
sin ((- sq) + z) is V11() real ext-real Element of REAL
sin (- sq) is V11() real ext-real Element of REAL
cos z is V11() real ext-real Element of REAL
(sin (- sq)) * (cos z) is V11() real ext-real Element of REAL
cos (- sq) is V11() real ext-real Element of REAL
sin z is V11() real ext-real Element of REAL
(cos (- sq)) * (sin z) is V11() real ext-real Element of REAL
((sin (- sq)) * (cos z)) + ((cos (- sq)) * (sin z)) is V11() real ext-real Element of REAL
- (sin sq) is V11() real ext-real Element of REAL
(- (sin sq)) * (cos z) is V11() real ext-real Element of REAL
((- (sin sq)) * (cos z)) + ((cos (- sq)) * (sin z)) is V11() real ext-real Element of REAL
(sin sq) * (cos z) is V11() real ext-real Element of REAL
- ((sin sq) * (cos z)) is V11() real ext-real Element of REAL
(cos sq) * (sin z) is V11() real ext-real Element of REAL
(- ((sin sq) * (cos z))) + ((cos sq) * (sin z)) is V11() real ext-real Element of REAL
- (sin z) is V11() real ext-real Element of REAL
(cos sq) * (- (sin z)) is V11() real ext-real Element of REAL
- ((cos sq) * (- (sin z))) is V11() real ext-real Element of REAL
(- ((sin sq) * (cos z))) + (- ((cos sq) * (- (sin z)))) is V11() real ext-real Element of REAL
((sin sq) * (cos z)) + ((cos sq) * (- (sin z))) is V11() real ext-real Element of REAL
((n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (cos z) is V11() real ext-real Element of REAL
((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (- (sin z)) is V11() real ext-real Element of REAL
(((n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (cos z)) + (((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (- (sin z))) is V11() real ext-real Element of REAL
(sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * ((((n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (cos z)) + (((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (- (sin z)))) is V11() real ext-real Element of REAL
(((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . p)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (cos z) is V11() real ext-real Element of REAL
(sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . q) is V11() real ext-real Element of REAL
((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . q)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . q)) * ((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) ") is V11() real ext-real set
(((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . q)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (- (sin z)) is V11() real ext-real Element of REAL
((((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . p)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (cos z)) + ((((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . q)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (- (sin z))) is V11() real ext-real Element of REAL
(n1 . p) * (cos z) is V11() real ext-real Element of REAL
(n1 . q) * (- (sin z)) is V11() real ext-real Element of REAL
((n1 . p) * (cos z)) + ((n1 . q) * (- (sin z))) is V11() real ext-real Element of REAL
- ((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) is V11() real ext-real Element of REAL
h is V11() real ext-real Element of REAL
h - sq is V11() real ext-real Element of REAL
- sq is V11() real ext-real set
h + (- sq) is V11() real ext-real set
z is V11() real ext-real Element of REAL
(TR,z,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,z,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
(Mx2Tran (TR,z,p,q)) . n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
((Mx2Tran (TR,z,p,q)) . n1) . p is V11() real ext-real Element of REAL
sin h is V11() real ext-real Element of REAL
sq + z is V11() real ext-real Element of REAL
sin (sq + z) is V11() real ext-real Element of REAL
cos z is V11() real ext-real Element of REAL
(sin sq) * (cos z) is V11() real ext-real Element of REAL
sin z is V11() real ext-real Element of REAL
(cos sq) * (sin z) is V11() real ext-real Element of REAL
((sin sq) * (cos z)) + ((cos sq) * (sin z)) is V11() real ext-real Element of REAL
((n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (cos z) is V11() real ext-real Element of REAL
- (sin z) is V11() real ext-real Element of REAL
((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (- (sin z)) is V11() real ext-real Element of REAL
(((n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (cos z)) + (((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (- (sin z))) is V11() real ext-real Element of REAL
(sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * ((((n1 . p) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (cos z)) + (((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (- (sin z)))) is V11() real ext-real Element of REAL
(((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . p)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (cos z) is V11() real ext-real Element of REAL
(sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . q) is V11() real ext-real Element of REAL
((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . q)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) is V11() real ext-real Element of REAL
((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . q)) * ((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) ") is V11() real ext-real set
(((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . q)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (- (sin z)) is V11() real ext-real Element of REAL
((((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . p)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (cos z)) + ((((sqrt (((n1 . p) ^2) + ((n1 . q) ^2))) * (n1 . q)) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) * (- (sin z))) is V11() real ext-real Element of REAL
(n1 . p) * (cos z) is V11() real ext-real Element of REAL
(n1 . q) * (- (sin z)) is V11() real ext-real Element of REAL
((n1 . p) * (cos z)) + ((n1 . q) * (- (sin z))) is V11() real ext-real Element of REAL
- ((n1 . q) / (sqrt (((n1 . p) ^2) + ((n1 . q) ^2)))) is V11() real ext-real Element of REAL
z is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
(TR,z,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,z,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
(Mx2Tran (TR,z,p,q)) . n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
((Mx2Tran (TR,z,p,q)) . n1) . p is V11() real ext-real Element of REAL
1. (F_Real,TR) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of TR,TR, the carrier of F_Real
Mx2Tran (1. (F_Real,TR)) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
id (TOP-REAL TR) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total quasi_total additive FinSequence-yielding Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
id the carrier of (TOP-REAL TR) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,n,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
(Mx2Tran (TR,n,p,q)) . n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
((Mx2Tran (TR,n,p,q)) . n1) . p is V11() real ext-real Element of REAL
(((Mx2Tran (TR,n,p,q)) . n1) . p) * (((Mx2Tran (TR,n,p,q)) . n1) . p) is V11() real ext-real Element of REAL
((Mx2Tran (TR,n,p,q)) . n1) . q is V11() real ext-real Element of REAL
(((Mx2Tran (TR,n,p,q)) . n1) . q) * (((Mx2Tran (TR,n,p,q)) . n1) . q) is V11() real ext-real Element of REAL
((((Mx2Tran (TR,n,p,q)) . n1) . p) * (((Mx2Tran (TR,n,p,q)) . n1) . p)) + ((((Mx2Tran (TR,n,p,q)) . n1) . q) * (((Mx2Tran (TR,n,p,q)) . n1) . q)) is V11() real ext-real Element of REAL
n1 . p is V11() real ext-real Element of REAL
(n1 . p) * (n1 . p) is V11() real ext-real Element of REAL
n1 . q is V11() real ext-real Element of REAL
(n1 . q) * (n1 . q) is V11() real ext-real Element of REAL
((n1 . p) * (n1 . p)) + ((n1 . q) * (n1 . q)) is V11() real ext-real Element of REAL
(n1 . p) ^2 is V11() real ext-real Element of REAL
(n1 . p) * (n1 . p) is V11() real ext-real set
(n1 . q) ^2 is V11() real ext-real Element of REAL
(n1 . q) * (n1 . q) is V11() real ext-real set
((n1 . p) ^2) + ((n1 . q) ^2) is V11() real ext-real Element of REAL
sin n is V11() real ext-real set
cos n is V11() real ext-real set
(cos n) * (cos n) is V11() real ext-real set
(sin n) * (sin n) is V11() real ext-real set
((cos n) * (cos n)) + ((sin n) * (sin n)) is V11() real ext-real set
(n1 . p) * (cos n) is V11() real ext-real Element of REAL
- (sin n) is V11() real ext-real set
(n1 . q) * (- (sin n)) is V11() real ext-real Element of REAL
((n1 . p) * (cos n)) + ((n1 . q) * (- (sin n))) is V11() real ext-real Element of REAL
((n1 . p) * (n1 . p)) * (cos n) is V11() real ext-real Element of REAL
(((n1 . p) * (n1 . p)) * (cos n)) * (cos n) is V11() real ext-real Element of REAL
2 * (n1 . p) is V11() real ext-real Element of REAL
(2 * (n1 . p)) * (n1 . q) is V11() real ext-real Element of REAL
((2 * (n1 . p)) * (n1 . q)) * (cos n) is V11() real ext-real Element of REAL
(((2 * (n1 . p)) * (n1 . q)) * (cos n)) * (sin n) is V11() real ext-real Element of REAL
((((n1 . p) * (n1 . p)) * (cos n)) * (cos n)) - ((((2 * (n1 . p)) * (n1 . q)) * (cos n)) * (sin n)) is V11() real ext-real Element of REAL
- ((((2 * (n1 . p)) * (n1 . q)) * (cos n)) * (sin n)) is V11() real ext-real set
((((n1 . p) * (n1 . p)) * (cos n)) * (cos n)) + (- ((((2 * (n1 . p)) * (n1 . q)) * (cos n)) * (sin n))) is V11() real ext-real set
((n1 . q) * (n1 . q)) * (sin n) is V11() real ext-real Element of REAL
(((n1 . q) * (n1 . q)) * (sin n)) * (sin n) is V11() real ext-real Element of REAL
(((((n1 . p) * (n1 . p)) * (cos n)) * (cos n)) - ((((2 * (n1 . p)) * (n1 . q)) * (cos n)) * (sin n))) + ((((n1 . q) * (n1 . q)) * (sin n)) * (sin n)) is V11() real ext-real Element of REAL
(n1 . p) * (sin n) is V11() real ext-real Element of REAL
(n1 . q) * (cos n) is V11() real ext-real Element of REAL
((n1 . p) * (sin n)) + ((n1 . q) * (cos n)) is V11() real ext-real Element of REAL
((n1 . p) * (n1 . p)) * (sin n) is V11() real ext-real Element of REAL
(((n1 . p) * (n1 . p)) * (sin n)) * (sin n) is V11() real ext-real Element of REAL
(n1 . p) * (n1 . q) is V11() real ext-real Element of REAL
2 * ((n1 . p) * (n1 . q)) is V11() real ext-real Element of REAL
(cos n) * (sin n) is V11() real ext-real set
(2 * ((n1 . p) * (n1 . q))) * ((cos n) * (sin n)) is V11() real ext-real Element of REAL
((((n1 . p) * (n1 . p)) * (sin n)) * (sin n)) + ((2 * ((n1 . p) * (n1 . q))) * ((cos n) * (sin n))) is V11() real ext-real Element of REAL
((n1 . q) * (n1 . q)) * ((cos n) * (cos n)) is V11() real ext-real Element of REAL
(((((n1 . p) * (n1 . p)) * (sin n)) * (sin n)) + ((2 * ((n1 . p) * (n1 . q))) * ((cos n) * (sin n)))) + (((n1 . q) * (n1 . q)) * ((cos n) * (cos n))) is V11() real ext-real Element of REAL
((n1 . p) * (n1 . p)) * (((cos n) * (cos n)) + ((sin n) * (sin n))) is V11() real ext-real Element of REAL
((n1 . q) * (n1 . q)) * (((cos n) * (cos n)) + ((sin n) * (sin n))) is V11() real ext-real Element of REAL
(((n1 . p) * (n1 . p)) * (((cos n) * (cos n)) + ((sin n) * (sin n)))) + (((n1 . q) * (n1 . q)) * (((cos n) * (cos n)) + ((sin n) * (sin n)))) is V11() real ext-real Element of REAL
n is V11() real ext-real set
n ^2 is V11() real ext-real set
n * n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
n1 . p is V11() real ext-real Element of REAL
(n1 . p) ^2 is V11() real ext-real Element of REAL
(n1 . p) * (n1 . p) is V11() real ext-real set
n1 . q is V11() real ext-real Element of REAL
(n1 . q) ^2 is V11() real ext-real Element of REAL
(n1 . q) * (n1 . q) is V11() real ext-real set
((n1 . p) ^2) + ((n1 . q) ^2) is V11() real ext-real Element of REAL
(((n1 . p) ^2) + ((n1 . q) ^2)) - (n ^2) is V11() real ext-real Element of REAL
- (n ^2) is V11() real ext-real set
(((n1 . p) ^2) + ((n1 . q) ^2)) + (- (n ^2)) is V11() real ext-real set
(((n1 . p) ^2) + ((n1 . q) ^2)) - {} is V11() real ext-real Element of REAL
- {} is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
(((n1 . p) ^2) + ((n1 . q) ^2)) + (- {}) is V11() real ext-real set
(n ^2) - (n ^2) is V11() real ext-real set
(n ^2) + (- (n ^2)) is V11() real ext-real set
sqrt ((((n1 . p) ^2) + ((n1 . q) ^2)) - (n ^2)) is V11() real ext-real Element of REAL
(sqrt ((((n1 . p) ^2) + ((n1 . q) ^2)) - (n ^2))) ^2 is V11() real ext-real Element of REAL
(sqrt ((((n1 . p) ^2) + ((n1 . q) ^2)) - (n ^2))) * (sqrt ((((n1 . p) ^2) + ((n1 . q) ^2)) - (n ^2))) is V11() real ext-real set
z is V11() real ext-real set
(TR,z,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,z,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
(Mx2Tran (TR,z,p,q)) . n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
((Mx2Tran (TR,z,p,q)) . n1) . p is V11() real ext-real Element of REAL
((Mx2Tran (TR,z,p,q)) . n1) . q is V11() real ext-real Element of REAL
(((Mx2Tran (TR,z,p,q)) . n1) . q) * (((Mx2Tran (TR,z,p,q)) . n1) . q) is V11() real ext-real Element of REAL
((sqrt ((((n1 . p) ^2) + ((n1 . q) ^2)) - (n ^2))) ^2) + ((((Mx2Tran (TR,z,p,q)) . n1) . q) * (((Mx2Tran (TR,z,p,q)) . n1) . q)) is V11() real ext-real Element of REAL
((((n1 . p) ^2) + ((n1 . q) ^2)) - (n ^2)) + ((((Mx2Tran (TR,z,p,q)) . n1) . q) * (((Mx2Tran (TR,z,p,q)) . n1) . q)) is V11() real ext-real Element of REAL
(((Mx2Tran (TR,z,p,q)) . n1) . q) ^2 is V11() real ext-real Element of REAL
(((Mx2Tran (TR,z,p,q)) . n1) . q) * (((Mx2Tran (TR,z,p,q)) . n1) . q) is V11() real ext-real set
- n is V11() real ext-real set
z + PI is V11() real ext-real Element of REAL
sq is V11() real ext-real Element of REAL
(TR,sq,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,sq,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
(Mx2Tran (TR,sq,p,q)) . n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
((Mx2Tran (TR,sq,p,q)) . n1) . q is V11() real ext-real Element of REAL
sin sq is V11() real ext-real Element of REAL
(n1 . p) * (sin sq) is V11() real ext-real Element of REAL
cos sq is V11() real ext-real Element of REAL
(n1 . q) * (cos sq) is V11() real ext-real Element of REAL
((n1 . p) * (sin sq)) + ((n1 . q) * (cos sq)) is V11() real ext-real Element of REAL
sin z is V11() real ext-real set
- (sin z) is V11() real ext-real set
(n1 . p) * (- (sin z)) is V11() real ext-real Element of REAL
((n1 . p) * (- (sin z))) + ((n1 . q) * (cos sq)) is V11() real ext-real Element of REAL
cos z is V11() real ext-real set
- (cos z) is V11() real ext-real set
(n1 . q) * (- (cos z)) is V11() real ext-real Element of REAL
((n1 . p) * (- (sin z))) + ((n1 . q) * (- (cos z))) is V11() real ext-real Element of REAL
(n1 . p) * (sin z) is V11() real ext-real Element of REAL
(n1 . q) * (cos z) is V11() real ext-real Element of REAL
((n1 . p) * (sin z)) + ((n1 . q) * (cos z)) is V11() real ext-real Element of REAL
- (((n1 . p) * (sin z)) + ((n1 . q) * (cos z))) is V11() real ext-real Element of REAL
- (((Mx2Tran (TR,z,p,q)) . n1) . q) is V11() real ext-real Element of REAL
- n is V11() real ext-real set
n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
{p,q} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,n,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
f is Relation-like Function-like set
dom (Mx2Tran (TR,n,p,q)) is non empty set
(Mx2Tran (TR,n,p,q)) . f is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
X is set
((Mx2Tran (TR,n,p,q)) . f) . X is V11() real ext-real set
f . X is set
((Mx2Tran (TR,n,p,q)) . f) . X is V11() real ext-real Element of REAL
z is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
len z is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom z is V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
Seg TR is V16() V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= TR ) } is set
(Mx2Tran (TR,n,p,q)) . z is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
len ((Mx2Tran (TR,n,p,q)) . z) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom ((Mx2Tran (TR,n,p,q)) . z) is V36() TR -element V155() V156() V157() V158() V159() V160() Element of bool NAT
((Mx2Tran (TR,n,p,q)) . z) . X is V11() real ext-real Element of REAL
((Mx2Tran (TR,n,p,q)) . z) . X is V11() real ext-real Element of REAL
z . X is V11() real ext-real Element of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Mx2Tran (p,n) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
n1 is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
|.n1.| is V11() real ext-real non negative Element of REAL
sqr n1 is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr n1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr n1)) is V11() real ext-real Element of REAL
(Mx2Tran (p,n)) . n1 is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
|.((Mx2Tran (p,n)) . n1).| is V11() real ext-real non negative Element of REAL
sqr ((Mx2Tran (p,n)) . n1) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr ((Mx2Tran (p,n)) . n1)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((Mx2Tran (p,n)) . n1))) is V11() real ext-real Element of REAL
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom n1 is V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
n1 . n is V11() real ext-real Element of REAL
- (n1 . n) is V11() real ext-real Element of REAL
n1 +* (n,(- (n1 . n))) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
sqr (n1 +* (n,(- (n1 . n)))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (n1 +* (n,(- (n1 . n))))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (n1 +* (n,(- (n1 . n)))))) is V11() real ext-real Element of REAL
(n1 . n) ^2 is V11() real ext-real Element of REAL
(n1 . n) * (n1 . n) is V11() real ext-real set
(Sum (sqr n1)) - ((n1 . n) ^2) is V11() real ext-real Element of REAL
- ((n1 . n) ^2) is V11() real ext-real set
(Sum (sqr n1)) + (- ((n1 . n) ^2)) is V11() real ext-real set
(- (n1 . n)) ^2 is V11() real ext-real Element of REAL
(- (n1 . n)) * (- (n1 . n)) is V11() real ext-real set
((Sum (sqr n1)) - ((n1 . n) ^2)) + ((- (n1 . n)) ^2) is V11() real ext-real Element of REAL
sqrt (((Sum (sqr n1)) - ((n1 . n) ^2)) + ((- (n1 . n)) ^2)) is V11() real ext-real Element of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total additive FinSequence-yielding Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
GFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions SubStr of GPFuncs the carrier of (TOP-REAL n)
GPFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions multMagma
the carrier of (GFuncs the carrier of (TOP-REAL n)) is non empty set
the carrier of (GFuncs the carrier of (TOP-REAL n)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
<*> the carrier of (GFuncs the carrier of (TOP-REAL n)) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
TR is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
Product TR is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
dom TR is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of bool NAT
1_ (GFuncs the carrier of (TOP-REAL n)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
the multF of (GFuncs the carrier of (TOP-REAL n)) is non empty Relation-like [: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like total quasi_total Element of bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):]
[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
[:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty set
the_unity_wrt the multF of (GFuncs the carrier of (TOP-REAL n)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR . n1 is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding Function-yielding V235() being_homeomorphism being_homeomorphism (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
p * q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
GFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions SubStr of GPFuncs the carrier of (TOP-REAL n)
GPFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions multMagma
the carrier of (GFuncs the carrier of (TOP-REAL n)) is non empty set
TR is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Product TR is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
dom TR is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
n1 is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Product n1 is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
dom n1 is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
the carrier of (GFuncs the carrier of (TOP-REAL n)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
n1 ^ TR is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
f is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
Product f is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
dom f is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(Product n1) * (Product TR) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
the multF of (GFuncs the carrier of (TOP-REAL n)) is non empty Relation-like [: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like total quasi_total Element of bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):]
[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
[:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty set
the multF of (GFuncs the carrier of (TOP-REAL n)) . ((Product n1),(Product TR)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
X is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
f . X is set
n1 . X is set
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(len n1) + z is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(len n1) + z is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
TR . z is set
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
f is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,n,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
|.n1.| is V11() real ext-real non negative Element of REAL
sqr n1 is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr n1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr n1)) is V11() real ext-real Element of REAL
(Mx2Tran (TR,n,p,q)) . n1 is Relation-like NAT -defined Function-like V36() TR -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL TR)
|.((Mx2Tran (TR,n,p,q)) . n1).| is V11() real ext-real non negative Element of REAL
sqr ((Mx2Tran (TR,n,p,q)) . n1) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr ((Mx2Tran (TR,n,p,q)) . n1)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((Mx2Tran (TR,n,p,q)) . n1))) is V11() real ext-real Element of REAL
@ n1 is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
@ (@ n1) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
p -' 1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
f is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
f | (p -' 1) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Seg (p -' 1) is V16() V36() p -' 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p -' 1 ) } is set
f | (Seg (p -' 1)) is Relation-like NAT -defined Seg (p -' 1) -defined NAT -defined REAL -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
q -' p is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(q -' p) -' 1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
f /^ p is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(f /^ p) | ((q -' p) -' 1) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Seg ((q -' p) -' 1) is V16() V36() (q -' p) -' 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= (q -' p) -' 1 ) } is set
(f /^ p) | (Seg ((q -' p) -' 1)) is Relation-like NAT -defined Seg ((q -' p) -' 1) -defined NAT -defined REAL -valued Function-like V36() FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
f /^ q is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
sin n is V11() real ext-real set
cos n is V11() real ext-real set
f . p is V11() real ext-real Element of REAL
f . q is V11() real ext-real Element of REAL
z is V11() real ext-real Element of REAL
<*z*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
[1,z] is set
{1,z} is non empty V36() V155() V156() V157() set
{{1,z},{1}} is non empty V36() V40() set
{[1,z]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
sqr <*z*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
z ^2 is V11() real ext-real Element of REAL
z * z is V11() real ext-real set
<*(z ^2)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(z ^2)] is set
{1,(z ^2)} is non empty V36() V155() V156() V157() set
{{1,(z ^2)},{1}} is non empty V36() V40() set
{[1,(z ^2)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
z * (cos n) is V11() real ext-real Element of REAL
z is V11() real ext-real Element of REAL
- (sin n) is V11() real ext-real set
z * (- (sin n)) is V11() real ext-real Element of REAL
(z * (cos n)) + (z * (- (sin n))) is V11() real ext-real Element of REAL
z * (sin n) is V11() real ext-real Element of REAL
z * (cos n) is V11() real ext-real Element of REAL
(z * (sin n)) + (z * (cos n)) is V11() real ext-real Element of REAL
<*((z * (cos n)) + (z * (- (sin n))))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,((z * (cos n)) + (z * (- (sin n))))] is set
{1,((z * (cos n)) + (z * (- (sin n))))} is non empty V36() V155() V156() V157() set
{{1,((z * (cos n)) + (z * (- (sin n))))},{1}} is non empty V36() V40() set
{[1,((z * (cos n)) + (z * (- (sin n))))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
sqr <*((z * (cos n)) + (z * (- (sin n))))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
((z * (cos n)) + (z * (- (sin n)))) ^2 is V11() real ext-real Element of REAL
((z * (cos n)) + (z * (- (sin n)))) * ((z * (cos n)) + (z * (- (sin n)))) is V11() real ext-real set
<*(((z * (cos n)) + (z * (- (sin n)))) ^2)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(((z * (cos n)) + (z * (- (sin n)))) ^2)] is set
{1,(((z * (cos n)) + (z * (- (sin n)))) ^2)} is non empty V36() V155() V156() V157() set
{{1,(((z * (cos n)) + (z * (- (sin n)))) ^2)},{1}} is non empty V36() V40() set
{[1,(((z * (cos n)) + (z * (- (sin n)))) ^2)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
<*((z * (sin n)) + (z * (cos n)))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,((z * (sin n)) + (z * (cos n)))] is set
{1,((z * (sin n)) + (z * (cos n)))} is non empty V36() V155() V156() V157() set
{{1,((z * (sin n)) + (z * (cos n)))},{1}} is non empty V36() V40() set
{[1,((z * (sin n)) + (z * (cos n)))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
sqr <*((z * (sin n)) + (z * (cos n)))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
((z * (sin n)) + (z * (cos n))) ^2 is V11() real ext-real Element of REAL
((z * (sin n)) + (z * (cos n))) * ((z * (sin n)) + (z * (cos n))) is V11() real ext-real set
<*(((z * (sin n)) + (z * (cos n))) ^2)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(((z * (sin n)) + (z * (cos n))) ^2)] is set
{1,(((z * (sin n)) + (z * (cos n))) ^2)} is non empty V36() V155() V156() V157() set
{{1,(((z * (sin n)) + (z * (cos n))) ^2)},{1}} is non empty V36() V40() set
{[1,(((z * (sin n)) + (z * (cos n))) ^2)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
<*z*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,z] is set
{1,z} is non empty V36() V155() V156() V157() set
{{1,z},{1}} is non empty V36() V40() set
{[1,z]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
sqr <*z*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
z ^2 is V11() real ext-real Element of REAL
z * z is V11() real ext-real set
<*(z ^2)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(z ^2)] is set
{1,(z ^2)} is non empty V36() V155() V156() V157() set
{{1,(z ^2)},{1}} is non empty V36() V40() set
{[1,(z ^2)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(f | (p -' 1)) ^ <*z*> is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((f | (p -' 1)) ^ <*z*>) ^ ((f /^ p) | ((q -' p) -' 1)) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
(((f | (p -' 1)) ^ <*z*>) ^ ((f /^ p) | ((q -' p) -' 1))) ^ <*z*> is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((((f | (p -' 1)) ^ <*z*>) ^ ((f /^ p) | ((q -' p) -' 1))) ^ <*z*>) ^ (f /^ q) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
sqr ((((f | (p -' 1)) ^ <*z*>) ^ ((f /^ p) | ((q -' p) -' 1))) ^ <*z*>) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
sqr (f /^ q) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(sqr ((((f | (p -' 1)) ^ <*z*>) ^ ((f /^ p) | ((q -' p) -' 1))) ^ <*z*>)) ^ (sqr (f /^ q)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
sqr (((f | (p -' 1)) ^ <*z*>) ^ ((f /^ p) | ((q -' p) -' 1))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(sqr (((f | (p -' 1)) ^ <*z*>) ^ ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*z*>) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((sqr (((f | (p -' 1)) ^ <*z*>) ^ ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*z*>)) ^ (sqr (f /^ q)) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
sqr ((f | (p -' 1)) ^ <*z*>) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
sqr ((f /^ p) | ((q -' p) -' 1)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(sqr ((f | (p -' 1)) ^ <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((sqr ((f | (p -' 1)) ^ <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*z*>) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
(((sqr ((f | (p -' 1)) ^ <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*z*>)) ^ (sqr (f /^ q)) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
sqr (f | (p -' 1)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(sqr (f | (p -' 1))) ^ (sqr <*z*>) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((sqr (f | (p -' 1))) ^ (sqr <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1))) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
(((sqr (f | (p -' 1))) ^ (sqr <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*z*>) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((((sqr (f | (p -' 1))) ^ (sqr <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*z*>)) ^ (sqr (f /^ q)) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
Sum ((((sqr (f | (p -' 1))) ^ (sqr <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*z*>)) is V11() real ext-real Element of REAL
Sum (sqr (f /^ q)) is V11() real ext-real Element of REAL
(Sum ((((sqr (f | (p -' 1))) ^ (sqr <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*z*>))) + (Sum (sqr (f /^ q))) is V11() real ext-real Element of REAL
Sum (((sqr (f | (p -' 1))) ^ (sqr <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) is V11() real ext-real Element of REAL
(Sum (((sqr (f | (p -' 1))) ^ (sqr <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1))))) + (z ^2) is V11() real ext-real Element of REAL
((Sum (((sqr (f | (p -' 1))) ^ (sqr <*z*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1))))) + (z ^2)) + (Sum (sqr (f /^ q))) is V11() real ext-real Element of REAL
Sum ((sqr (f | (p -' 1))) ^ (sqr <*z*>)) is V11() real ext-real Element of REAL
Sum (sqr ((f /^ p) | ((q -' p) -' 1))) is V11() real ext-real Element of REAL
(Sum ((sqr (f | (p -' 1))) ^ (sqr <*z*>))) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1)))) is V11() real ext-real Element of REAL
((Sum ((sqr (f | (p -' 1))) ^ (sqr <*z*>))) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1))))) + (z ^2) is V11() real ext-real Element of REAL
(((Sum ((sqr (f | (p -' 1))) ^ (sqr <*z*>))) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1))))) + (z ^2)) + (Sum (sqr (f /^ q))) is V11() real ext-real Element of REAL
Sum (sqr (f | (p -' 1))) is V11() real ext-real Element of REAL
(Sum (sqr (f | (p -' 1)))) + (z ^2) is V11() real ext-real Element of REAL
((Sum (sqr (f | (p -' 1)))) + (z ^2)) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1)))) is V11() real ext-real Element of REAL
(((Sum (sqr (f | (p -' 1)))) + (z ^2)) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1))))) + (z ^2) is V11() real ext-real Element of REAL
((((Sum (sqr (f | (p -' 1)))) + (z ^2)) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1))))) + (z ^2)) + (Sum (sqr (f /^ q))) is V11() real ext-real Element of REAL
(cos n) * (cos n) is V11() real ext-real set
(sin n) * (sin n) is V11() real ext-real set
((cos n) * (cos n)) + ((sin n) * (sin n)) is V11() real ext-real set
(((z * (cos n)) + (z * (- (sin n)))) ^2) + (((z * (sin n)) + (z * (cos n))) ^2) is V11() real ext-real Element of REAL
z * z is V11() real ext-real Element of REAL
(z * z) * (((cos n) * (cos n)) + ((sin n) * (sin n))) is V11() real ext-real Element of REAL
z * z is V11() real ext-real Element of REAL
(z * z) * (((cos n) * (cos n)) + ((sin n) * (sin n))) is V11() real ext-real Element of REAL
((z * z) * (((cos n) * (cos n)) + ((sin n) * (sin n)))) + ((z * z) * (((cos n) * (cos n)) + ((sin n) * (sin n)))) is V11() real ext-real Element of REAL
(z ^2) + (z ^2) is V11() real ext-real Element of REAL
(f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*> is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>) ^ ((f /^ p) | ((q -' p) -' 1)) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
(((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>) ^ ((f /^ p) | ((q -' p) -' 1))) ^ <*((z * (sin n)) + (z * (cos n)))*> is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>) ^ ((f /^ p) | ((q -' p) -' 1))) ^ <*((z * (sin n)) + (z * (cos n)))*>) ^ (f /^ q) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
sqr ((((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>) ^ ((f /^ p) | ((q -' p) -' 1))) ^ <*((z * (sin n)) + (z * (cos n)))*>) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(sqr ((((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>) ^ ((f /^ p) | ((q -' p) -' 1))) ^ <*((z * (sin n)) + (z * (cos n)))*>)) ^ (sqr (f /^ q)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
sqr (((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>) ^ ((f /^ p) | ((q -' p) -' 1))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(sqr (((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>) ^ ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*((z * (sin n)) + (z * (cos n)))*>) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((sqr (((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>) ^ ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*((z * (sin n)) + (z * (cos n)))*>)) ^ (sqr (f /^ q)) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
sqr ((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(sqr ((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((sqr ((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*((z * (sin n)) + (z * (cos n)))*>) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
(((sqr ((f | (p -' 1)) ^ <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*((z * (sin n)) + (z * (cos n)))*>)) ^ (sqr (f /^ q)) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
(sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1))) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
(((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*((z * (sin n)) + (z * (cos n)))*>) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
((((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*((z * (sin n)) + (z * (cos n)))*>)) ^ (sqr (f /^ q)) is non empty Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL *
Sum ((((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*((z * (sin n)) + (z * (cos n)))*>)) is V11() real ext-real Element of REAL
(Sum ((((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) ^ (sqr <*((z * (sin n)) + (z * (cos n)))*>))) + (Sum (sqr (f /^ q))) is V11() real ext-real Element of REAL
Sum (((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1)))) is V11() real ext-real Element of REAL
(Sum (((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1))))) + (((z * (sin n)) + (z * (cos n))) ^2) is V11() real ext-real Element of REAL
((Sum (((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>)) ^ (sqr ((f /^ p) | ((q -' p) -' 1))))) + (((z * (sin n)) + (z * (cos n))) ^2)) + (Sum (sqr (f /^ q))) is V11() real ext-real Element of REAL
Sum ((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>)) is V11() real ext-real Element of REAL
(Sum ((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>))) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1)))) is V11() real ext-real Element of REAL
((Sum ((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>))) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1))))) + (((z * (sin n)) + (z * (cos n))) ^2) is V11() real ext-real Element of REAL
(((Sum ((sqr (f | (p -' 1))) ^ (sqr <*((z * (cos n)) + (z * (- (sin n))))*>))) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1))))) + (((z * (sin n)) + (z * (cos n))) ^2)) + (Sum (sqr (f /^ q))) is V11() real ext-real Element of REAL
(Sum (sqr (f | (p -' 1)))) + (((z * (cos n)) + (z * (- (sin n)))) ^2) is V11() real ext-real Element of REAL
((Sum (sqr (f | (p -' 1)))) + (((z * (cos n)) + (z * (- (sin n)))) ^2)) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1)))) is V11() real ext-real Element of REAL
(((Sum (sqr (f | (p -' 1)))) + (((z * (cos n)) + (z * (- (sin n)))) ^2)) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1))))) + (((z * (sin n)) + (z * (cos n))) ^2) is V11() real ext-real Element of REAL
((((Sum (sqr (f | (p -' 1)))) + (((z * (cos n)) + (z * (- (sin n)))) ^2)) + (Sum (sqr ((f /^ p) | ((q -' p) -' 1))))) + (((z * (sin n)) + (z * (cos n))) ^2)) + (Sum (sqr (f /^ q))) is V11() real ext-real Element of REAL
n is V11() real ext-real set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TR is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(TR,n,p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of TR,TR, the carrier of F_Real
Mx2Tran (TR,n,p,q) is non empty Relation-like the carrier of (TOP-REAL TR) -defined the carrier of (TOP-REAL TR) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):]
TOP-REAL TR is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL TR) is non empty set
[: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL TR), the carrier of (TOP-REAL TR):] is non empty set
GFuncs the carrier of (TOP-REAL TR) is non empty strict unital associative constituted-Functions SubStr of GPFuncs the carrier of (TOP-REAL TR)
GPFuncs the carrier of (TOP-REAL TR) is non empty strict unital associative constituted-Functions multMagma
the carrier of (GFuncs the carrier of (TOP-REAL TR)) is non empty set
1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL TR)) is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL TR))
X is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL TR))
<*X*> is non empty trivial Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL TR)) -valued Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like finite-support Function-yielding V235() V282() Element of 1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL TR))
[1,X] is set
{1,X} is non empty V36() set
{{1,X},{1}} is non empty V36() V40() set
{[1,X]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
z is non empty trivial Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL TR)) -valued Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like finite-support Function-yielding V235() V282() Element of 1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL TR))
Product z is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL TR))
dom z is non empty trivial V36() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
z . fp is Relation-like Function-like set
{1} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
z . 1 is Relation-like Function-like set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
TR is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
q * TR is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n1 is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
|.n1.| is V11() real ext-real non negative Element of REAL
sqr n1 is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr n1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr n1)) is V11() real ext-real Element of REAL
(q * TR) . n1 is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
|.((q * TR) . n1).| is V11() real ext-real non negative Element of REAL
sqr ((q * TR) . n1) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr ((q * TR) . n1)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((q * TR) . n1))) is V11() real ext-real Element of REAL
dom (q * TR) is non empty set
TR . n1 is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
q . (TR . n1) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
|.(q . (TR . n1)).| is V11() real ext-real non negative Element of REAL
sqr (q . (TR . n1)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (q . (TR . n1))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (q . (TR . n1)))) is V11() real ext-real Element of REAL
|.(TR . n1).| is V11() real ext-real non negative Element of REAL
sqr (TR . n1) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (TR . n1)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (TR . n1))) is V11() real ext-real Element of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
GFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions SubStr of GPFuncs the carrier of (TOP-REAL n)
GPFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions multMagma
TR is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
the carrier of (GFuncs the carrier of (TOP-REAL n)) is non empty set
n1 is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Product n1 is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
dom n1 is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n1 | f is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg f is V16() V36() f -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
n1 | (Seg f) is Relation-like NAT -defined Seg f -defined NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSubsequence-like finite-support set
Product (n1 | f) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
f + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n1 | (f + 1) is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg (f + 1) is non empty V16() V36() f + 1 -element f + 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= f + 1 ) } is set
n1 | (Seg (f + 1)) is Relation-like NAT -defined Seg (f + 1) -defined NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSubsequence-like finite-support set
Product (n1 | (f + 1)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
n1 . (f + 1) is set
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
fpz is V11() real ext-real set
(n,fpz,fp,z) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
Mx2Tran (n,fpz,fp,z) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
h is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
<*h*> is non empty trivial Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like finite-support Function-yielding V235() V282() Element of 1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL n))
1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL n)) is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
[1,h] is set
{1,h} is non empty V36() set
{{1,h},{1}} is non empty V36() V40() set
{[1,h]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(n1 | f) ^ <*h*> is non empty Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
the carrier of (GFuncs the carrier of (TOP-REAL n)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
sq is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Product (n1 | f)) * h is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
the multF of (GFuncs the carrier of (TOP-REAL n)) is non empty Relation-like [: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like total quasi_total Element of bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):]
[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
[:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty set
the multF of (GFuncs the carrier of (TOP-REAL n)) . ((Product (n1 | f)),h) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
z is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran (n,fpz,fp,z)) * z is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n1 | (len n1) is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg (len n1) is V16() V36() len n1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= len n1 ) } is set
n1 | (Seg (len n1)) is Relation-like NAT -defined Seg (len n1) -defined NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSubsequence-like finite-support set
n1 | {} is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg {} is empty ordinal natural V11() real ext-real non positive non negative V16() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
n1 | (Seg {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined Seg {} -defined NAT -defined RAT -valued the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
Product (n1 | {}) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding Function-yielding V235() being_homeomorphism being_homeomorphism (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
f is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
|.f.| is V11() real ext-real non negative Element of REAL
sqr f is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr f) is V11() real ext-real Element of REAL
sqrt (Sum (sqr f)) is V11() real ext-real Element of REAL
(id (TOP-REAL n)) . f is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
|.((id (TOP-REAL n)) . f).| is V11() real ext-real non negative Element of REAL
sqr ((id (TOP-REAL n)) . f) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr ((id (TOP-REAL n)) . f)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((id (TOP-REAL n)) . f))) is V11() real ext-real Element of REAL
f is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
X is V11() real ext-real set
z is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
X * z is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
X * z is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(id (TOP-REAL n)) . (X * z) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(id (TOP-REAL n)) . z is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
X * ((id (TOP-REAL n)) . z) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
X * ((id (TOP-REAL n)) . z) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
<*> the carrier of (GFuncs the carrier of (TOP-REAL n)) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
the carrier of (GFuncs the carrier of (TOP-REAL n)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
1_ (GFuncs the carrier of (TOP-REAL n)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
the multF of (GFuncs the carrier of (TOP-REAL n)) is non empty Relation-like [: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like total quasi_total Element of bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):]
[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
[:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty set
the_unity_wrt the multF of (GFuncs the carrier of (TOP-REAL n)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
p /" is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
GFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions SubStr of GPFuncs the carrier of (TOP-REAL n)
GPFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions multMagma
the carrier of (GFuncs the carrier of (TOP-REAL n)) is non empty set
n1 is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Product n1 is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
dom n1 is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
f + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
len z is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Product z is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
dom z is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
fp is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
fp /" is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
z | f is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg f is V16() V36() f -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
z | (Seg f) is Relation-like NAT -defined Seg f -defined NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSubsequence-like finite-support set
Product (z | f) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
z . (f + 1) is set
h is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
sq is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
z is V11() real ext-real set
(n,z,h,sq) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
Mx2Tran (n,z,h,sq) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
k is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
<*k*> is non empty trivial Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like finite-support Function-yielding V235() V282() Element of 1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL n))
1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL n)) is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
[1,k] is set
{1,k} is non empty V36() set
{{1,k},{1}} is non empty V36() V40() set
{[1,k]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(z | f) ^ <*k*> is non empty Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
the carrier of (GFuncs the carrier of (TOP-REAL n)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
(Product (z | f)) * k is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
the multF of (GFuncs the carrier of (TOP-REAL n)) is non empty Relation-like [: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like total quasi_total Element of bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):]
[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
[:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty set
the multF of (GFuncs the carrier of (TOP-REAL n)) . ((Product (z | f)),k) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
fpz is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran (n,z,h,sq)) * fpz is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom (z | f) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
gf is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(z | f) . gf is set
z . gf is set
Det (n,z,h,sq) is V11() real ext-real Element of the carrier of F_Real
(n,z,h,sq) ~ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
- z is V11() real ext-real set
(n,(- z),h,sq) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
(Mx2Tran (n,z,h,sq)) " is Relation-like Function-like set
Mx2Tran (n,(- z),h,sq) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
rng (Mx2Tran (n,z,h,sq)) is non empty set
[#] (TOP-REAL n) is non empty non proper Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
dom (Mx2Tran (n,z,h,sq)) is non empty set
(Mx2Tran (n,z,h,sq)) /" is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
len (z | f) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
fpz /" is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom fpz is non empty set
rng fpz is non empty set
(fpz /") * ((Mx2Tran (n,z,h,sq)) /") is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
f is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
len f is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Product f is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
dom f is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
X is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
X /" is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
<*> the carrier of (GFuncs the carrier of (TOP-REAL n)) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
the carrier of (GFuncs the carrier of (TOP-REAL n)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
1_ (GFuncs the carrier of (TOP-REAL n)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
the multF of (GFuncs the carrier of (TOP-REAL n)) is non empty Relation-like [: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like total quasi_total Element of bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):]
[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
[:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty set
the_unity_wrt the multF of (GFuncs the carrier of (TOP-REAL n)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
rng X is non empty set
[#] (TOP-REAL n) is non empty non proper Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
X " is Relation-like Function-like set
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
f is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
len f is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Product f is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
X is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom f is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
X /" is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
p * q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
TR is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n -VectSp_over F_Real is non empty right_complementable strict V139( F_Real ) V140( F_Real ) V141( F_Real ) V142( F_Real ) V189() V190() V191() V260( F_Real ) VectSpStr over F_Real
the carrier of (n -VectSp_over F_Real) is non empty set
1. (F_Real,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
MX2FinS (1. (F_Real,n)) is Relation-like NAT -defined the carrier of (n -VectSp_over F_Real) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (n -VectSp_over F_Real)
REAL n is non empty functional FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
[: the carrier of (n -VectSp_over F_Real), the carrier of (n -VectSp_over F_Real):] is non empty Relation-like set
bool [: the carrier of (n -VectSp_over F_Real), the carrier of (n -VectSp_over F_Real):] is non empty set
n -tuples_on the carrier of F_Real is non empty functional FinSequence-membered FinSequenceSet of the carrier of F_Real
X is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
z is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
f is non empty Relation-like the carrier of (n -VectSp_over F_Real) -defined the carrier of (n -VectSp_over F_Real) -valued Function-like total quasi_total Element of bool [: the carrier of (n -VectSp_over F_Real), the carrier of (n -VectSp_over F_Real):]
f . X is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
f . z is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
fpz is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() n -element FinSequence-like FinSubsequence-like finite-support Element of n -tuples_on the carrier of F_Real
@ fpz is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
@ (@ fpz) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
h is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() n -element FinSequence-like FinSubsequence-like finite-support Element of n -tuples_on the carrier of F_Real
@ h is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
@ (@ h) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
fp is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() n -element FinSequence-like FinSubsequence-like finite-support Element of n -tuples_on the carrier of F_Real
@ fp is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
@ (@ fp) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
z is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() n -element FinSequence-like FinSubsequence-like finite-support Element of n -tuples_on the carrier of F_Real
@ z is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
@ (@ z) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
X + z is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
the addF of (n -VectSp_over F_Real) is non empty Relation-like [: the carrier of (n -VectSp_over F_Real), the carrier of (n -VectSp_over F_Real):] -defined the carrier of (n -VectSp_over F_Real) -valued Function-like total quasi_total Element of bool [:[: the carrier of (n -VectSp_over F_Real), the carrier of (n -VectSp_over F_Real):], the carrier of (n -VectSp_over F_Real):]
[:[: the carrier of (n -VectSp_over F_Real), the carrier of (n -VectSp_over F_Real):], the carrier of (n -VectSp_over F_Real):] is non empty Relation-like set
bool [:[: the carrier of (n -VectSp_over F_Real), the carrier of (n -VectSp_over F_Real):], the carrier of (n -VectSp_over F_Real):] is non empty set
the addF of (n -VectSp_over F_Real) . (X,z) is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
fp + z is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
the addF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the addF of F_Real,fp,z) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
sq is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
z is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
sq + z is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) is non empty Relation-like [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total Element of bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):]
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
the addF of (TOP-REAL n) . (sq,z) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
sq + z is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
f . (X + z) is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
p . sq is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p . z is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(p . sq) + (p . z) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) . ((p . sq),(p . z)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(p . sq) + (p . z) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
fpz + h is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
K721( the carrier of F_Real, the carrier of F_Real, the carrier of F_Real, the addF of F_Real,fpz,h) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
(f . X) + (f . z) is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
the addF of (n -VectSp_over F_Real) . ((f . X),(f . z)) is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
n1 is Relation-like NAT -defined the carrier of (n -VectSp_over F_Real) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support OrdBasis of n -VectSp_over F_Real
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
AutMt (f,n1,n1) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of len n1, len n1, the carrier of F_Real
X is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran X is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
fp is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
f . fp is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
z is V11() real ext-real Element of the carrier of F_Real
z * fp is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
the lmult of (n -VectSp_over F_Real) is non empty Relation-like [: the carrier of F_Real, the carrier of (n -VectSp_over F_Real):] -defined the carrier of (n -VectSp_over F_Real) -valued Function-like total quasi_total Element of bool [:[: the carrier of F_Real, the carrier of (n -VectSp_over F_Real):], the carrier of (n -VectSp_over F_Real):]
[: the carrier of F_Real, the carrier of (n -VectSp_over F_Real):] is non empty Relation-like set
[:[: the carrier of F_Real, the carrier of (n -VectSp_over F_Real):], the carrier of (n -VectSp_over F_Real):] is non empty Relation-like set
bool [:[: the carrier of F_Real, the carrier of (n -VectSp_over F_Real):], the carrier of (n -VectSp_over F_Real):] is non empty set
the lmult of (n -VectSp_over F_Real) . (z,fp) is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
fpz is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() n -element FinSequence-like FinSubsequence-like finite-support Element of n -tuples_on the carrier of F_Real
z * fpz is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
z multfield is non empty Relation-like the carrier of F_Real -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of F_Real, the carrier of F_Real:]
bool [: the carrier of F_Real, the carrier of F_Real:] is non empty set
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
id the carrier of F_Real is non empty Relation-like the carrier of F_Real -defined the carrier of F_Real -valued Function-like one-to-one total quasi_total complex-yielding ext-real-valued real-valued V149() non-decreasing Element of bool [: the carrier of F_Real, the carrier of F_Real:]
the multF of F_Real [;] (z,(id the carrier of F_Real)) is non empty Relation-like the carrier of F_Real -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Element of bool [: the carrier of F_Real, the carrier of F_Real:]
K724( the carrier of F_Real, the carrier of F_Real,fpz,(z multfield)) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
z is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
z * z is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
z * z is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
f . (z * fp) is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
p . z is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
z * (p . z) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
z * (p . z) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() n -element FinSequence-like FinSubsequence-like finite-support Element of n -tuples_on the carrier of F_Real
z * h is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
K724( the carrier of F_Real, the carrier of F_Real,h,(z multfield)) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
z * (f . fp) is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
the lmult of (n -VectSp_over F_Real) . (z,(f . fp)) is Relation-like Function-like Element of the carrier of (n -VectSp_over F_Real)
Mx2Tran ((AutMt (f,n1,n1)),n1,n1) is non empty Relation-like the carrier of (n -VectSp_over F_Real) -defined the carrier of (n -VectSp_over F_Real) -valued Function-like total quasi_total Element of bool [: the carrier of (n -VectSp_over F_Real), the carrier of (n -VectSp_over F_Real):]
q is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
TR is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran TR is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
p * q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,(p * q)) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(n,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(n,q) * (n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
width (n,p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
width (n,q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (n,q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Mx2Tran ((n,q) * (n,p)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Mx2Tran (n,q) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Mx2Tran (n,p) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran (n,p)) * (Mx2Tran (n,q)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
p * (Mx2Tran (n,q)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n is set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL q is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL q) is non empty set
Seg q is V16() V36() q -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= q ) } is set
[: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):] is non empty set
n /\ (Seg q) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
card (n /\ (Seg q)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
TR is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
{p} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Seg f is V16() V36() f -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
n /\ (Seg f) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(n /\ (Seg f)) \/ {p} is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
card ((n /\ (Seg f)) \/ {p}) is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
f + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Seg (f + 1) is non empty V16() V36() f + 1 -element f + 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= f + 1 ) } is set
n /\ (Seg (f + 1)) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(n /\ (Seg (f + 1))) \/ {p} is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
card ((n /\ (Seg (f + 1))) \/ {p}) is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
{(f + 1)} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
(Seg f) \/ {(f + 1)} is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
{(f + 1),p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
((n /\ (Seg f)) \/ {p}) \/ {(f + 1),p} is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
n \/ {(f + 1)} is non empty set
{(f + 1)} \/ {p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{p} \/ {(f + 1)} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{p} \/ ({p} \/ {(f + 1)}) is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
(n /\ (Seg f)) \/ ({p} \/ ({p} \/ {(f + 1)})) is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
{p} \/ {p} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
({p} \/ {p}) \/ {(f + 1)} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
(n /\ (Seg f)) \/ (({p} \/ {p}) \/ {(f + 1)}) is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(n /\ (Seg f)) \/ {(f + 1)} is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
((n /\ (Seg f)) \/ {(f + 1)}) \/ {p} is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
z is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
z . TR is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
(z . TR) . p is V11() real ext-real Element of REAL
z is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
z . TR is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
(z . TR) . p is V11() real ext-real Element of REAL
(Seg (f + 1)) \/ {(f + 1)} is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
{(f + 1)} \/ {(f + 1)} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
(Seg f) \/ ({(f + 1)} \/ {(f + 1)}) is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
n \/ {p} is non empty set
(Seg f) \/ {p} is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(n \/ {p}) /\ ((Seg f) \/ {p}) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(z . TR) . z is V11() real ext-real Element of REAL
n /\ {(f + 1)} is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(n /\ (Seg f)) \/ (n /\ {(f + 1)}) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(n /\ (Seg f)) \/ {} is V36() V155() V156() V157() V158() V159() V160() set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(z . TR) . z is V11() real ext-real Element of REAL
(z . TR) . (f + 1) is V11() real ext-real Element of REAL
((z . TR) . (f + 1)) ^2 is V11() real ext-real Element of REAL
((z . TR) . (f + 1)) * ((z . TR) . (f + 1)) is V11() real ext-real set
((z . TR) . p) ^2 is V11() real ext-real Element of REAL
((z . TR) . p) * ((z . TR) . p) is V11() real ext-real set
(((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2) is V11() real ext-real Element of REAL
sqrt ((((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2)) is V11() real ext-real Element of REAL
(sqrt ((((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2))) ^2 is V11() real ext-real Element of REAL
(sqrt ((((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2))) * (sqrt ((((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2))) is V11() real ext-real set
h is V11() real ext-real set
(q,h,(f + 1),p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of q,q, the carrier of F_Real
Mx2Tran (q,h,(f + 1),p) is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
(Mx2Tran (q,h,(f + 1),p)) . (z . TR) is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
((Mx2Tran (q,h,(f + 1),p)) . (z . TR)) . p is V11() real ext-real Element of REAL
sq is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
sq * z is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued the carrier of (TOP-REAL q) -valued the carrier of (TOP-REAL q) -valued Function-like total total total quasi_total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
z is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
z . TR is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
(z . TR) . p is V11() real ext-real Element of REAL
dom z is non empty set
dom sq is non empty set
sq . (z . TR) is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(z . TR) . z is V11() real ext-real Element of REAL
(z . TR) . z is V11() real ext-real Element of REAL
(sq . (z . TR)) . z is V11() real ext-real Element of REAL
((sq . (z . TR)) . z) * ((sq . (z . TR)) . z) is V11() real ext-real Element of REAL
(((sq . (z . TR)) . z) * ((sq . (z . TR)) . z)) + ((((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2)) is V11() real ext-real Element of REAL
(z . TR) . (f + 1) is V11() real ext-real Element of REAL
((z . TR) . (f + 1)) ^2 is V11() real ext-real Element of REAL
((z . TR) . (f + 1)) * ((z . TR) . (f + 1)) is V11() real ext-real set
((z . TR) . p) ^2 is V11() real ext-real Element of REAL
((z . TR) . p) * ((z . TR) . p) is V11() real ext-real set
(((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2) is V11() real ext-real Element of REAL
sqrt ((((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2)) is V11() real ext-real Element of REAL
(sqrt ((((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2))) ^2 is V11() real ext-real Element of REAL
(sqrt ((((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2))) * (sqrt ((((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2))) is V11() real ext-real set
h is V11() real ext-real set
(q,h,p,(f + 1)) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of q,q, the carrier of F_Real
Mx2Tran (q,h,p,(f + 1)) is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
(Mx2Tran (q,h,p,(f + 1))) . (z . TR) is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
((Mx2Tran (q,h,p,(f + 1))) . (z . TR)) . p is V11() real ext-real Element of REAL
sq is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
sq * z is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued the carrier of (TOP-REAL q) -valued the carrier of (TOP-REAL q) -valued Function-like total total total quasi_total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
z is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
z . TR is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
(z . TR) . p is V11() real ext-real Element of REAL
{p,(f + 1)} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
dom z is non empty set
sq . (z . TR) is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(z . TR) . z is V11() real ext-real Element of REAL
dom sq is non empty set
(z . TR) . z is V11() real ext-real Element of REAL
(sq . (z . TR)) . z is V11() real ext-real Element of REAL
((sq . (z . TR)) . z) * ((sq . (z . TR)) . z) is V11() real ext-real Element of REAL
(((sq . (z . TR)) . z) * ((sq . (z . TR)) . z)) + ((((z . TR) . (f + 1)) ^2) + (((z . TR) . p) ^2)) is V11() real ext-real Element of REAL
Seg {} is empty ordinal natural V11() real ext-real non positive non negative V16() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
n /\ (Seg {}) is Relation-like V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(n /\ (Seg {})) \/ {p} is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
card ((n /\ (Seg {})) \/ {p}) is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
id (TOP-REAL q) is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued the carrier of (TOP-REAL q) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
id the carrier of (TOP-REAL q) is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
f is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
f . TR is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
(f . TR) . p is V11() real ext-real Element of REAL
X is Relation-like Function-like set
dom f is non empty set
f . X is Relation-like Function-like set
z is set
(f . X) . z is set
X . z is set
X is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(f . TR) . X is V11() real ext-real Element of REAL
(n /\ (Seg q)) \/ {p} is non empty V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
card ((n /\ (Seg q)) \/ {p}) is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
f is non empty Relation-like the carrier of (TOP-REAL q) -defined the carrier of (TOP-REAL q) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (q) (q) Element of bool [: the carrier of (TOP-REAL q), the carrier of (TOP-REAL q):]
f . TR is Relation-like NAT -defined Function-like V36() q -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL q)
(f . TR) . p is V11() real ext-real Element of REAL
X is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(f . TR) . X is V11() real ext-real Element of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
bool the carrier of (TOP-REAL n) is non empty set
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
TR is Element of bool the carrier of (TOP-REAL n)
p | TR is Relation-like the carrier of (TOP-REAL n) -defined TR -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id TR is Relation-like TR -defined TR -valued Function-like one-to-one total quasi_total Element of bool [:TR,TR:]
[:TR,TR:] is Relation-like set
bool [:TR,TR:] is non empty set
Lin TR is non empty right_complementable strict V189() V190() V191() V192() V193() V194() V195() M25( TOP-REAL n)
the carrier of (Lin TR) is non empty set
p | (Lin TR) is non empty Relation-like the carrier of (Lin TR) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total Element of bool [: the carrier of (Lin TR), the carrier of (TOP-REAL n):]
[: the carrier of (Lin TR), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (Lin TR), the carrier of (TOP-REAL n):] is non empty set
id (Lin TR) is non empty Relation-like the carrier of (Lin TR) -defined the carrier of (Lin TR) -valued Function-like total quasi_total additive Element of bool [: the carrier of (Lin TR), the carrier of (Lin TR):]
[: the carrier of (Lin TR), the carrier of (Lin TR):] is non empty Relation-like set
bool [: the carrier of (Lin TR), the carrier of (Lin TR):] is non empty set
id the carrier of (Lin TR) is non empty Relation-like the carrier of (Lin TR) -defined the carrier of (Lin TR) -valued Function-like one-to-one total quasi_total Element of bool [: the carrier of (Lin TR), the carrier of (Lin TR):]
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
X is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TR
Carrier X is V36() Element of bool the carrier of (TOP-REAL n)
card (Carrier X) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
Sum X is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p . (Sum X) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
z is set
fp is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{fp} is non empty trivial functional V36() V40() 1 -element set
z is Element of bool the carrier of (TOP-REAL n)
X . fp is V11() real ext-real Element of REAL
fpz is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of z
fpz . fp is V11() real ext-real Element of REAL
Carrier fpz is V36() Element of bool the carrier of (TOP-REAL n)
{fp} \/ (Carrier X) is non empty V36() set
X - fpz is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
Carrier (X - fpz) is V36() Element of bool the carrier of (TOP-REAL n)
(X - fpz) . fp is V11() real ext-real Element of REAL
(X . fp) - (fpz . fp) is V11() real ext-real Element of REAL
- (fpz . fp) is V11() real ext-real set
(X . fp) + (- (fpz . fp)) is V11() real ext-real set
card (Carrier (X - fpz)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
ZeroLC (TOP-REAL n) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
- fpz is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
(- fpz) - (- fpz) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
- (- fpz) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
(- fpz) + (- (- fpz)) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
(- fpz) + fpz is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
X + ((- fpz) + fpz) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
X + (- fpz) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
(X + (- fpz)) + fpz is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
(X - fpz) + fpz is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TOP-REAL n
Sum (X - fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
Sum fpz is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(Sum (X - fpz)) + (Sum fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) is non empty Relation-like [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total Element of bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):]
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
the addF of (TOP-REAL n) . ((Sum (X - fpz)),(Sum fpz)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(Sum (X - fpz)) + (Sum fpz) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(p | TR) . fp is Relation-like Function-like set
p . fp is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p . (Sum (X - fpz)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(X . fp) * fp is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(X . fp) * fp is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
p . (Sum fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p | the carrier of (Lin TR) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (Lin TR) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
X is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TR
Carrier X is V36() Element of bool the carrier of (TOP-REAL n)
card (Carrier X) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
Sum X is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p . (Sum X) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{} the carrier of (TOP-REAL n) is empty proper ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of bool the carrier of (TOP-REAL n)
0. (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element V61( TOP-REAL n) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p . (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{} * (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{} * (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
p . ({} * (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{} * (p . (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{} * (p . (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
X is set
(p | (Lin TR)) . X is set
(id (Lin TR)) . X is set
z is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TR
Sum z is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
Carrier z is V36() Element of bool the carrier of (TOP-REAL n)
card (Carrier z) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
fp is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued Linear_Combination of TR
Carrier fp is V36() Element of bool the carrier of (TOP-REAL n)
card (Carrier fp) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
Sum fp is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p . (Sum fp) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p . X is Relation-like Function-like set
dom (p | (Lin TR)) is non empty set
dom (id (Lin TR)) is non empty set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
bool the carrier of (TOP-REAL n) is non empty set
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
q is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
q . p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
n1 is Element of bool the carrier of (TOP-REAL n)
q | n1 is Relation-like the carrier of (TOP-REAL n) -defined n1 -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id n1 is Relation-like n1 -defined n1 -valued Function-like one-to-one total quasi_total Element of bool [:n1,n1:]
[:n1,n1:] is Relation-like set
bool [:n1,n1:] is non empty set
Lin n1 is non empty right_complementable strict V189() V190() V191() V192() V193() V194() V195() M25( TOP-REAL n)
0* n is Relation-like NAT -defined REAL -valued Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL n
REAL n is non empty functional FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of n -tuples_on REAL
the carrier of (Lin n1) is non empty set
q | (Lin n1) is non empty Relation-like the carrier of (Lin n1) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total Element of bool [: the carrier of (Lin n1), the carrier of (TOP-REAL n):]
[: the carrier of (Lin n1), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (Lin n1), the carrier of (TOP-REAL n):] is non empty set
id (Lin n1) is non empty Relation-like the carrier of (Lin n1) -defined the carrier of (Lin n1) -valued Function-like total quasi_total additive Element of bool [: the carrier of (Lin n1), the carrier of (Lin n1):]
[: the carrier of (Lin n1), the carrier of (Lin n1):] is non empty Relation-like set
bool [: the carrier of (Lin n1), the carrier of (Lin n1):] is non empty set
id the carrier of (Lin n1) is non empty Relation-like the carrier of (Lin n1) -defined the carrier of (Lin n1) -valued Function-like one-to-one total quasi_total Element of bool [: the carrier of (Lin n1), the carrier of (Lin n1):]
len (0* n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (0* n) is V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Base_FinSeq (n,z) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(q . p) . z is V11() real ext-real Element of REAL
p . z is V11() real ext-real Element of REAL
(0* n) +* (z,1) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
q | the carrier of (Lin n1) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (Lin n1) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom ((0* n) +* (z,1)) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
len ((0* n) +* (z,1)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
@ ((0* n) +* (z,1)) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
@ (@ ((0* n) +* (z,1))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
h . z is V11() real ext-real Element of REAL
q . h is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(q . h) . z is V11() real ext-real Element of REAL
fpz is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(h . z) * fpz is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(h . z) * fpz is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h - ((h . z) * fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
- ((h . z) * fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
- ((h . z) * fpz) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h + (- ((h . z) * fpz)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) is non empty Relation-like [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total Element of bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):]
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
the addF of (TOP-REAL n) . (h,(- ((h . z) * fpz))) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
h + (- ((h . z) * fpz)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h - ((h . z) * fpz) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
q . (h - ((h . z) * fpz)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
q . ((h . z) * fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(q . (h - ((h . z) * fpz))) + (q . ((h . z) * fpz)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) . ((q . (h - ((h . z) * fpz))),(q . ((h . z) * fpz))) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(q . (h - ((h . z) * fpz))) + (q . ((h . z) * fpz)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(h - ((h . z) * fpz)) + ((h . z) * fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) . ((h - ((h . z) * fpz)),((h . z) * fpz)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(h - ((h . z) * fpz)) + ((h . z) * fpz) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
q . ((h - ((h . z) * fpz)) + ((h . z) * fpz)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
len (q . (h - ((h . z) * fpz))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (q . ((h . z) * fpz)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
|.(q . ((h - ((h . z) * fpz)) + ((h . z) * fpz))).| is V11() real ext-real non negative Element of REAL
sqr (q . ((h - ((h . z) * fpz)) + ((h . z) * fpz))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (q . ((h - ((h . z) * fpz)) + ((h . z) * fpz)))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (q . ((h - ((h . z) * fpz)) + ((h . z) * fpz))))) is V11() real ext-real Element of REAL
|.(q . ((h - ((h . z) * fpz)) + ((h . z) * fpz))).| ^2 is V11() real ext-real Element of REAL
|.(q . ((h - ((h . z) * fpz)) + ((h . z) * fpz))).| * |.(q . ((h - ((h . z) * fpz)) + ((h . z) * fpz))).| is V11() real ext-real non negative set
|.(q . (h - ((h . z) * fpz))).| is V11() real ext-real non negative Element of REAL
sqr (q . (h - ((h . z) * fpz))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (q . (h - ((h . z) * fpz)))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (q . (h - ((h . z) * fpz))))) is V11() real ext-real Element of REAL
|.(q . (h - ((h . z) * fpz))).| ^2 is V11() real ext-real Element of REAL
|.(q . (h - ((h . z) * fpz))).| * |.(q . (h - ((h . z) * fpz))).| is V11() real ext-real non negative set
|((q . ((h . z) * fpz)),(q . (h - ((h . z) * fpz))))| is V11() real ext-real Element of REAL
2 * |((q . ((h . z) * fpz)),(q . (h - ((h . z) * fpz))))| is V11() real ext-real Element of REAL
(|.(q . (h - ((h . z) * fpz))).| ^2) + (2 * |((q . ((h . z) * fpz)),(q . (h - ((h . z) * fpz))))|) is V11() real ext-real Element of REAL
|.(q . ((h . z) * fpz)).| is V11() real ext-real non negative Element of REAL
sqr (q . ((h . z) * fpz)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (q . ((h . z) * fpz))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (q . ((h . z) * fpz)))) is V11() real ext-real Element of REAL
|.(q . ((h . z) * fpz)).| ^2 is V11() real ext-real Element of REAL
|.(q . ((h . z) * fpz)).| * |.(q . ((h . z) * fpz)).| is V11() real ext-real non negative set
((|.(q . (h - ((h . z) * fpz))).| ^2) + (2 * |((q . ((h . z) * fpz)),(q . (h - ((h . z) * fpz))))|)) + (|.(q . ((h . z) * fpz)).| ^2) is V11() real ext-real Element of REAL
n |-> (h . z) is Relation-like NAT -defined REAL -valued Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of n -tuples_on REAL
(n |-> (h . z)) . z is V11() real ext-real Element of REAL
(h . z) * ((0* n) +* (z,1)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
mlt ((n |-> (h . z)),((0* n) +* (z,1))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(h . z) * 1 is V11() real ext-real Element of REAL
(0* n) +* (z,((h . z) * 1)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
(0* n) +* (z,(h . z)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
len (q . h) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (q . h) is V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
len (h - ((h . z) * fpz)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (h - ((h . z) * fpz)) is V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
(h - ((h . z) * fpz)) . z is V11() real ext-real Element of REAL
((h . z) * fpz) . z is V11() real ext-real Element of REAL
(h . z) - (((h . z) * fpz) . z) is V11() real ext-real Element of REAL
- (((h . z) * fpz) . z) is V11() real ext-real set
(h . z) + (- (((h . z) * fpz) . z)) is V11() real ext-real set
len ((h . z) * fpz) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
|.((h - ((h . z) * fpz)) + ((h . z) * fpz)).| is V11() real ext-real non negative Element of REAL
sqr ((h - ((h . z) * fpz)) + ((h . z) * fpz)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr ((h - ((h . z) * fpz)) + ((h . z) * fpz))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((h - ((h . z) * fpz)) + ((h . z) * fpz)))) is V11() real ext-real Element of REAL
|.((h - ((h . z) * fpz)) + ((h . z) * fpz)).| ^2 is V11() real ext-real Element of REAL
|.((h - ((h . z) * fpz)) + ((h . z) * fpz)).| * |.((h - ((h . z) * fpz)) + ((h . z) * fpz)).| is V11() real ext-real non negative set
|.(h - ((h . z) * fpz)).| is V11() real ext-real non negative Element of REAL
sqr (h - ((h . z) * fpz)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (h - ((h . z) * fpz))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (h - ((h . z) * fpz)))) is V11() real ext-real Element of REAL
|.(h - ((h . z) * fpz)).| ^2 is V11() real ext-real Element of REAL
|.(h - ((h . z) * fpz)).| * |.(h - ((h . z) * fpz)).| is V11() real ext-real non negative set
|(((h . z) * fpz),(h - ((h . z) * fpz)))| is V11() real ext-real Element of REAL
2 * |(((h . z) * fpz),(h - ((h . z) * fpz)))| is V11() real ext-real Element of REAL
(|.(h - ((h . z) * fpz)).| ^2) + (2 * |(((h . z) * fpz),(h - ((h . z) * fpz)))|) is V11() real ext-real Element of REAL
|.((h . z) * fpz).| is V11() real ext-real non negative Element of REAL
sqr ((h . z) * fpz) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr ((h . z) * fpz)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((h . z) * fpz))) is V11() real ext-real Element of REAL
|.((h . z) * fpz).| ^2 is V11() real ext-real Element of REAL
|.((h . z) * fpz).| * |.((h . z) * fpz).| is V11() real ext-real non negative set
((|.(h - ((h . z) * fpz)).| ^2) + (2 * |(((h . z) * fpz),(h - ((h . z) * fpz)))|)) + (|.((h . z) * fpz).| ^2) is V11() real ext-real Element of REAL
(q | (Lin n1)) . ((h . z) * fpz) is set
(q . (h - ((h . z) * fpz))) . z is V11() real ext-real Element of REAL
(h . z) * ((q . (h - ((h . z) * fpz))) . z) is V11() real ext-real Element of REAL
(h . z) * ((h - ((h . z) * fpz)) . z) is V11() real ext-real Element of REAL
((h . z) * fpz) - ((h . z) * fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
((h . z) * fpz) + (- ((h . z) * fpz)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) . (((h . z) * fpz),(- ((h . z) * fpz))) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
((h . z) * fpz) + (- ((h . z) * fpz)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
((h . z) * fpz) - ((h . z) * fpz) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h - (((h . z) * fpz) - ((h . z) * fpz)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
- (((h . z) * fpz) - ((h . z) * fpz)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
- (((h . z) * fpz) - ((h . z) * fpz)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h + (- (((h . z) * fpz) - ((h . z) * fpz))) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) . (h,(- (((h . z) * fpz) - ((h . z) * fpz)))) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
h + (- (((h . z) * fpz) - ((h . z) * fpz))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h - (((h . z) * fpz) - ((h . z) * fpz)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
0. (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element V61( TOP-REAL n) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
h - (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
- (0. (TOP-REAL n)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
- (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h + (- (0. (TOP-REAL n))) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) . (h,(- (0. (TOP-REAL n)))) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
h + (- (0. (TOP-REAL n))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h - (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(q . ((h . z) * fpz)) . z is V11() real ext-real Element of REAL
((q . (h - ((h . z) * fpz))) . z) + ((q . ((h . z) * fpz)) . z) is V11() real ext-real Element of REAL
((q . (h - ((h . z) * fpz))) . z) + (h . z) is V11() real ext-real Element of REAL
len (q . p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
fpz is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(q . p) + fpz is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) is non empty Relation-like [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total Element of bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):]
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
the addF of (TOP-REAL n) . ((q . p),fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(q . p) + fpz is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
|.((q . p) + fpz).| is V11() real ext-real non negative Element of REAL
sqr ((q . p) + fpz) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr ((q . p) + fpz)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((q . p) + fpz))) is V11() real ext-real Element of REAL
|.((q . p) + fpz).| ^2 is V11() real ext-real Element of REAL
|.((q . p) + fpz).| * |.((q . p) + fpz).| is V11() real ext-real non negative set
|.(q . p).| is V11() real ext-real non negative Element of REAL
sqr (q . p) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (q . p)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (q . p))) is V11() real ext-real Element of REAL
|.(q . p).| ^2 is V11() real ext-real Element of REAL
|.(q . p).| * |.(q . p).| is V11() real ext-real non negative set
|(fpz,(q . p))| is V11() real ext-real Element of REAL
2 * |(fpz,(q . p))| is V11() real ext-real Element of REAL
(|.(q . p).| ^2) + (2 * |(fpz,(q . p))|) is V11() real ext-real Element of REAL
|.fpz.| is V11() real ext-real non negative Element of REAL
sqr fpz is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr fpz) is V11() real ext-real Element of REAL
sqrt (Sum (sqr fpz)) is V11() real ext-real Element of REAL
|.fpz.| ^2 is V11() real ext-real Element of REAL
|.fpz.| * |.fpz.| is V11() real ext-real non negative set
((|.(q . p).| ^2) + (2 * |(fpz,(q . p))|)) + (|.fpz.| ^2) is V11() real ext-real Element of REAL
len p is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
p + fpz is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) . (p,fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
p + fpz is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
|.(p + fpz).| is V11() real ext-real non negative Element of REAL
sqr (p + fpz) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (p + fpz)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (p + fpz))) is V11() real ext-real Element of REAL
|.(p + fpz).| ^2 is V11() real ext-real Element of REAL
|.(p + fpz).| * |.(p + fpz).| is V11() real ext-real non negative set
|.p.| is V11() real ext-real non negative Element of REAL
sqr p is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr p) is V11() real ext-real Element of REAL
sqrt (Sum (sqr p)) is V11() real ext-real Element of REAL
|.p.| ^2 is V11() real ext-real Element of REAL
|.p.| * |.p.| is V11() real ext-real non negative set
|(fpz,p)| is V11() real ext-real Element of REAL
2 * |(fpz,p)| is V11() real ext-real Element of REAL
(|.p.| ^2) + (2 * |(fpz,p)|) is V11() real ext-real Element of REAL
((|.p.| ^2) + (2 * |(fpz,p)|)) + (|.fpz.| ^2) is V11() real ext-real Element of REAL
(q | (Lin n1)) . fpz is set
q . fpz is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
q . (p + fpz) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
|.(q . (p + fpz)).| is V11() real ext-real non negative Element of REAL
sqr (q . (p + fpz)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (q . (p + fpz))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (q . (p + fpz)))) is V11() real ext-real Element of REAL
1 * (p . z) is V11() real ext-real Element of REAL
1 * ((q . p) . z) is V11() real ext-real Element of REAL
n is set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
n /\ (Seg p) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
n1 is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() (p) Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
n1 . q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
sqr q is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
sqr (n1 . q) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr q) is V11() real ext-real Element of REAL
Sum (sqr (n1 . q)) is V11() real ext-real Element of REAL
|.(n1 . q).| is V11() real ext-real non negative Element of REAL
sqrt (Sum (sqr (n1 . q))) is V11() real ext-real Element of REAL
|.q.| is V11() real ext-real non negative Element of REAL
sqrt (Sum (sqr q)) is V11() real ext-real Element of REAL
len q is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
@ q is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
@ (@ q) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
len (sqr q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (n1 . q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
@ (n1 . q) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
@ (@ (n1 . q)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
len (sqr (n1 . q)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
p -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
dom n1 is non empty set
z is Relation-like NAT -defined REAL -valued Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of p -tuples_on REAL
fp is Relation-like NAT -defined REAL -valued Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of p -tuples_on REAL
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
z . z is V11() real ext-real Element of REAL
fp . z is V11() real ext-real Element of REAL
q . z is V11() real ext-real Element of REAL
(q . z) ^2 is V11() real ext-real Element of REAL
(q . z) * (q . z) is V11() real ext-real set
(n1 . q) . z is V11() real ext-real Element of REAL
((n1 . q) . z) ^2 is V11() real ext-real Element of REAL
((n1 . q) . z) * ((n1 . q) . z) is V11() real ext-real set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q . z is V11() real ext-real Element of REAL
(n1 . q) . z is V11() real ext-real Element of REAL
z . z is V11() real ext-real Element of REAL
(q . z) ^2 is V11() real ext-real Element of REAL
(q . z) * (q . z) is V11() real ext-real set
fp . z is V11() real ext-real Element of REAL
((n1 . q) . z) ^2 is V11() real ext-real Element of REAL
((n1 . q) . z) * ((n1 . q) . z) is V11() real ext-real set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
q . n is V11() real ext-real Element of REAL
- (q . n) is V11() real ext-real Element of REAL
q +* (n,(- (q . n))) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
{n} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
card (Seg p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
card {n} is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
n1 is set
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
len (q +* (n,(- (q . n)))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len q is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom q is V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
(q . n) * (q . n) is V11() real ext-real Element of REAL
q . f is V11() real ext-real Element of REAL
(q . f) * (q . f) is V11() real ext-real Element of REAL
{} ^2 is V11() real ext-real set
{} * {} is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
0* {} is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of REAL {}
REAL {} is non empty functional FinSequence-membered FinSequenceSet of REAL
{} -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
{} |-> 0 is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined REAL -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of {} -tuples_on REAL
(q . n) ^2 is V11() real ext-real Element of REAL
(q . n) * (q . n) is V11() real ext-real set
(q . f) ^2 is V11() real ext-real Element of REAL
(q . f) * (q . f) is V11() real ext-real set
((q . n) ^2) + ((q . f) ^2) is V11() real ext-real Element of REAL
z is V11() real ext-real set
(p,z,n,f) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Mx2Tran (p,z,n,f) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
(Mx2Tran (p,z,n,f)) . q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
((Mx2Tran (p,z,n,f)) . q) . n is V11() real ext-real Element of REAL
sin z is V11() real ext-real set
cos z is V11() real ext-real set
(q . n) * (cos z) is V11() real ext-real Element of REAL
- (sin z) is V11() real ext-real set
(q . f) * (- (sin z)) is V11() real ext-real Element of REAL
((q . n) * (cos z)) + ((q . f) * (- (sin z))) is V11() real ext-real Element of REAL
z + z is V11() real ext-real set
(p,(z + z),n,f) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Mx2Tran (p,(z + z),n,f) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
fpz is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (p) (p) Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
fpz . q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
cos (z + z) is V11() real ext-real set
(cos z) * (cos z) is V11() real ext-real set
(sin z) * (sin z) is V11() real ext-real set
((cos z) * (cos z)) - ((sin z) * (sin z)) is V11() real ext-real set
- ((sin z) * (sin z)) is V11() real ext-real set
((cos z) * (cos z)) + (- ((sin z) * (sin z))) is V11() real ext-real set
sin (z + z) is V11() real ext-real set
(sin z) * (cos z) is V11() real ext-real set
((sin z) * (cos z)) + ((sin z) * (cos z)) is V11() real ext-real set
{n,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
sq is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(q +* (n,(- (q . n)))) . sq is V11() real ext-real Element of REAL
(fpz . q) . sq is V11() real ext-real Element of REAL
(q . n) * 1 is V11() real ext-real Element of REAL
- ((q . n) * 1) is V11() real ext-real Element of REAL
((sin z) * (sin z)) + ((cos z) * (cos z)) is V11() real ext-real set
(q . n) * (((sin z) * (sin z)) + ((cos z) * (cos z))) is V11() real ext-real Element of REAL
- ((q . n) * (((sin z) * (sin z)) + ((cos z) * (cos z)))) is V11() real ext-real Element of REAL
(q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z))) is V11() real ext-real Element of REAL
((q . n) * (cos z)) * (cos z) is V11() real ext-real Element of REAL
((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) - (((q . n) * (cos z)) * (cos z)) is V11() real ext-real Element of REAL
- (((q . n) * (cos z)) * (cos z)) is V11() real ext-real set
((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) + (- (((q . n) * (cos z)) * (cos z))) is V11() real ext-real set
(((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) - (((q . n) * (cos z)) * (cos z))) - (((q . n) * (cos z)) * (cos z)) is V11() real ext-real Element of REAL
(((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) - (((q . n) * (cos z)) * (cos z))) + (- (((q . n) * (cos z)) * (cos z))) is V11() real ext-real set
(q . f) * (sin z) is V11() real ext-real Element of REAL
((q . f) * (sin z)) * (cos z) is V11() real ext-real Element of REAL
((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) - (((q . f) * (sin z)) * (cos z)) is V11() real ext-real Element of REAL
- (((q . f) * (sin z)) * (cos z)) is V11() real ext-real set
((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) + (- (((q . f) * (sin z)) * (cos z))) is V11() real ext-real set
(((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) - (((q . f) * (sin z)) * (cos z))) - (((q . f) * (sin z)) * (cos z)) is V11() real ext-real Element of REAL
(((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) - (((q . f) * (sin z)) * (cos z))) + (- (((q . f) * (sin z)) * (cos z))) is V11() real ext-real set
- (((sin z) * (cos z)) + ((sin z) * (cos z))) is V11() real ext-real set
(q . f) * (- (((sin z) * (cos z)) + ((sin z) * (cos z)))) is V11() real ext-real Element of REAL
((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) + ((q . f) * (- (((sin z) * (cos z)) + ((sin z) * (cos z))))) is V11() real ext-real Element of REAL
(q . f) * 1 is V11() real ext-real Element of REAL
((sin z) * (sin z)) + ((cos z) * (cos z)) is V11() real ext-real set
(q . f) * (((sin z) * (sin z)) + ((cos z) * (cos z))) is V11() real ext-real Element of REAL
(q . f) * (sin z) is V11() real ext-real Element of REAL
((q . f) * (sin z)) * (sin z) is V11() real ext-real Element of REAL
(((q . f) * (sin z)) * (sin z)) + (((q . f) * (sin z)) * (sin z)) is V11() real ext-real Element of REAL
(q . f) * (((cos z) * (cos z)) - ((sin z) * (sin z))) is V11() real ext-real Element of REAL
((((q . f) * (sin z)) * (sin z)) + (((q . f) * (sin z)) * (sin z))) + ((q . f) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) is V11() real ext-real Element of REAL
((q . n) * (cos z)) * (sin z) is V11() real ext-real Element of REAL
(((q . n) * (cos z)) * (sin z)) + (((q . n) * (cos z)) * (sin z)) is V11() real ext-real Element of REAL
((((q . n) * (cos z)) * (sin z)) + (((q . n) * (cos z)) * (sin z))) + ((q . f) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) is V11() real ext-real Element of REAL
(q . n) * (((sin z) * (cos z)) + ((sin z) * (cos z))) is V11() real ext-real Element of REAL
((q . n) * (((sin z) * (cos z)) + ((sin z) * (cos z)))) + ((q . f) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) is V11() real ext-real Element of REAL
dom fpz is non empty set
q . sq is V11() real ext-real Element of REAL
len (fpz . q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is V11() real ext-real set
(p,z,f,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Mx2Tran (p,z,f,n) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
(Mx2Tran (p,z,f,n)) . q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
((Mx2Tran (p,z,f,n)) . q) . n is V11() real ext-real Element of REAL
sin z is V11() real ext-real set
cos z is V11() real ext-real set
(q . f) * (sin z) is V11() real ext-real Element of REAL
(q . n) * (cos z) is V11() real ext-real Element of REAL
((q . f) * (sin z)) + ((q . n) * (cos z)) is V11() real ext-real Element of REAL
z + z is V11() real ext-real set
(p,(z + z),f,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
Mx2Tran (p,(z + z),f,n) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
fpz is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (p) (p) Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
fpz . q is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
cos (z + z) is V11() real ext-real set
(cos z) * (cos z) is V11() real ext-real set
(sin z) * (sin z) is V11() real ext-real set
((cos z) * (cos z)) - ((sin z) * (sin z)) is V11() real ext-real set
- ((sin z) * (sin z)) is V11() real ext-real set
((cos z) * (cos z)) + (- ((sin z) * (sin z))) is V11() real ext-real set
sin (z + z) is V11() real ext-real set
(sin z) * (cos z) is V11() real ext-real set
((sin z) * (cos z)) + ((sin z) * (cos z)) is V11() real ext-real set
{n,f} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
sq is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(q +* (n,(- (q . n)))) . sq is V11() real ext-real Element of REAL
(fpz . q) . sq is V11() real ext-real Element of REAL
(q . n) * 1 is V11() real ext-real Element of REAL
- ((q . n) * 1) is V11() real ext-real Element of REAL
((sin z) * (sin z)) + ((cos z) * (cos z)) is V11() real ext-real set
(q . n) * (((sin z) * (sin z)) + ((cos z) * (cos z))) is V11() real ext-real Element of REAL
- ((q . n) * (((sin z) * (sin z)) + ((cos z) * (cos z)))) is V11() real ext-real Element of REAL
((q . n) * (cos z)) * (cos z) is V11() real ext-real Element of REAL
- (((q . n) * (cos z)) * (cos z)) is V11() real ext-real Element of REAL
- ((q . n) * (cos z)) is V11() real ext-real Element of REAL
(- ((q . n) * (cos z))) * (cos z) is V11() real ext-real Element of REAL
(- (((q . n) * (cos z)) * (cos z))) + ((- ((q . n) * (cos z))) * (cos z)) is V11() real ext-real Element of REAL
(q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z))) is V11() real ext-real Element of REAL
((- (((q . n) * (cos z)) * (cos z))) + ((- ((q . n) * (cos z))) * (cos z))) + ((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) is V11() real ext-real Element of REAL
((q . f) * (sin z)) * (cos z) is V11() real ext-real Element of REAL
(((q . f) * (sin z)) * (cos z)) + (((q . f) * (sin z)) * (cos z)) is V11() real ext-real Element of REAL
((((q . f) * (sin z)) * (cos z)) + (((q . f) * (sin z)) * (cos z))) + ((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) is V11() real ext-real Element of REAL
(q . f) * (((sin z) * (cos z)) + ((sin z) * (cos z))) is V11() real ext-real Element of REAL
((q . f) * (((sin z) * (cos z)) + ((sin z) * (cos z)))) + ((q . n) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) is V11() real ext-real Element of REAL
(q . f) * 1 is V11() real ext-real Element of REAL
((sin z) * (sin z)) + ((cos z) * (cos z)) is V11() real ext-real set
(q . f) * (((sin z) * (sin z)) + ((cos z) * (cos z))) is V11() real ext-real Element of REAL
(q . f) * (((cos z) * (cos z)) - ((sin z) * (sin z))) is V11() real ext-real Element of REAL
((q . f) * (sin z)) * (sin z) is V11() real ext-real Element of REAL
((q . f) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) + (((q . f) * (sin z)) * (sin z)) is V11() real ext-real Element of REAL
(((q . f) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) + (((q . f) * (sin z)) * (sin z))) + (((q . f) * (sin z)) * (sin z)) is V11() real ext-real Element of REAL
- ((q . n) * (cos z)) is V11() real ext-real Element of REAL
(- ((q . n) * (cos z))) * (sin z) is V11() real ext-real Element of REAL
((q . f) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) + ((- ((q . n) * (cos z))) * (sin z)) is V11() real ext-real Element of REAL
(((q . f) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) + ((- ((q . n) * (cos z))) * (sin z))) + ((- ((q . n) * (cos z))) * (sin z)) is V11() real ext-real Element of REAL
- (((sin z) * (cos z)) + ((sin z) * (cos z))) is V11() real ext-real set
(q . n) * (- (((sin z) * (cos z)) + ((sin z) * (cos z)))) is V11() real ext-real Element of REAL
((q . f) * (((cos z) * (cos z)) - ((sin z) * (sin z)))) + ((q . n) * (- (((sin z) * (cos z)) + ((sin z) * (cos z))))) is V11() real ext-real Element of REAL
dom fpz is non empty set
q . sq is V11() real ext-real Element of REAL
len (fpz . q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n is set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
q is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Base_FinSeq (p,q) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
TR is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() (p) Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
TR . (Base_FinSeq (p,q)) is Relation-like Function-like set
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
n /\ (Seg p) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
(Base_FinSeq (p,q)) . f is V11() real ext-real Element of REAL
len (Base_FinSeq (p,q)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
{n} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
1. (F_Real,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
q is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
(p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
{ (Base_FinSeq (p,b₁)) where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
f is set
X is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Base_FinSeq (p,X) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
len (Base_FinSeq (p,X)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
bool the carrier of (TOP-REAL p) is non empty set
(n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
Mx2Tran (n,p) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
id (TOP-REAL p) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued the carrier of (TOP-REAL p) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (p) (p) Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
id the carrier of (TOP-REAL p) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
Mx2Tran (1. (F_Real,p)) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
(p,(id (TOP-REAL p))) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
dom q is non empty set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
z is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
q . z is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
(q . z) . fp is V11() real ext-real Element of REAL
z . fp is V11() real ext-real Element of REAL
len (q . z) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len z is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
Base_FinSeq (p,n) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
f is Element of bool the carrier of (TOP-REAL p)
z is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
0* p is Relation-like NAT -defined REAL -valued Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL p
REAL p is non empty functional FinSequence-membered FinSequenceSet of REAL
p -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
p |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of p -tuples_on REAL
(0* p) +* (n,1) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
|.z.| is V11() real ext-real non negative Element of REAL
sqr z is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr z) is V11() real ext-real Element of REAL
sqrt (Sum (sqr z)) is V11() real ext-real Element of REAL
abs 1 is V11() real ext-real Element of REAL
q . z is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
(q . z) . n is V11() real ext-real Element of REAL
(0* p) +* (n,((q . z) . n)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
len (0* p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(q . z) . z is V11() real ext-real Element of REAL
((0* p) +* (n,((q . z) . n))) . z is V11() real ext-real Element of REAL
dom (0* p) is V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
(0* p) . z is V11() real ext-real Element of REAL
z . z is V11() real ext-real Element of REAL
len ((0* p) +* (n,((q . z) . n))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (q . z) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
|.((0* p) +* (n,((q . z) . n))).| is V11() real ext-real non negative Element of REAL
sqr ((0* p) +* (n,((q . z) . n))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr ((0* p) +* (n,((q . z) . n)))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((0* p) +* (n,((q . z) . n))))) is V11() real ext-real Element of REAL
abs ((q . z) . n) is V11() real ext-real Element of REAL
id f is Relation-like f -defined f -valued Function-like one-to-one total quasi_total Element of bool [:f,f:]
[:f,f:] is Relation-like set
bool [:f,f:] is non empty set
z is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() (p) Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
z | f is Relation-like the carrier of (TOP-REAL p) -defined f -defined the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
dom (id (TOP-REAL p)) is non empty set
fpz is set
(id (TOP-REAL p)) . fpz is Relation-like Function-like set
z . fpz is Relation-like Function-like set
h is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
z . h is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
len h is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (z . h) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
Base_FinSeq (p,z) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Lin f is non empty right_complementable strict V189() V190() V191() V192() V193() V194() V195() M25( TOP-REAL p)
h . z is V11() real ext-real Element of REAL
(z . h) . z is V11() real ext-real Element of REAL
dom z is non empty set
q | f is Relation-like the carrier of (TOP-REAL p) -defined f -defined the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
dom (q | f) is set
z is set
(q | f) . z is Relation-like Function-like set
(id f) . z is set
fpz is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Base_FinSeq (p,fpz) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
q . z is Relation-like Function-like set
dom (id f) is set
Mx2Tran (p,n) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
(Mx2Tran (p,n)) * q is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued the carrier of (TOP-REAL p) -valued the carrier of (TOP-REAL p) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
fpz is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() (p) Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
dom fpz is non empty set
fpz | f is Relation-like the carrier of (TOP-REAL p) -defined f -defined the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
dom (fpz | f) is set
{n} \/ {n} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
h is set
(fpz | f) . h is Relation-like Function-like set
(id f) . h is set
sq is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Base_FinSeq (p,sq) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
fpz . h is Relation-like Function-like set
fpz . z is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(fpz . z) . z is V11() real ext-real Element of REAL
z . z is V11() real ext-real Element of REAL
(Mx2Tran (p,n)) . (q . z) is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
((Mx2Tran (p,n)) . (q . z)) . z is V11() real ext-real Element of REAL
(q . z) . z is V11() real ext-real Element of REAL
- ((q . z) . z) is V11() real ext-real Element of REAL
- (- 1) is non empty V11() real ext-real positive non negative V85() Element of REAL
(Mx2Tran (p,n)) . (q . z) is Relation-like NAT -defined Function-like V36() p -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL p)
((Mx2Tran (p,n)) . (q . z)) . z is V11() real ext-real Element of REAL
(q . z) . z is V11() real ext-real Element of REAL
len (fpz . z) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len z is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (id f) is set
dom (Mx2Tran (p,n)) is non empty set
[#] (TOP-REAL p) is non empty non proper Element of bool the carrier of (TOP-REAL p)
rng (Mx2Tran (p,n)) is non empty set
rng q is non empty set
sq is set
(Mx2Tran (p,n)) . sq is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
fpz . ((Mx2Tran (p,n)) . sq) is Relation-like Function-like set
q . ((Mx2Tran (p,n)) . sq) is Relation-like Function-like set
(Mx2Tran (p,n)) . (q . ((Mx2Tran (p,n)) . sq)) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
(Mx2Tran (p,n)) " is Relation-like Function-like set
Mx2Tran (p,q) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
Det (p,n) is V11() real ext-real Element of the carrier of F_Real
(p,n) ~ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
{n} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom p is non empty set
rng p is non empty set
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
f is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
f * p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom (id the carrier of (TOP-REAL n)) is non empty set
dom f is non empty set
X is Relation-like Function-like set
f . X is Relation-like Function-like set
z is set
(f . X) . z is set
X . z is set
0. (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element V61( TOP-REAL n) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{(0. (TOP-REAL n))} is non empty trivial functional V36() V40() 1 -element set
rng (id the carrier of (TOP-REAL n)) is non empty set
n - 1 is V11() real ext-real V85() Element of REAL
n + (- 1) is V11() real ext-real V85() set
bool the carrier of (TOP-REAL n) is non empty set
{ (Base_FinSeq (n,b₁)) where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= a₁ ) } is set
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
{ (Base_FinSeq (n,b₁)) where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
z is set
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Base_FinSeq (n,fp) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
len (Base_FinSeq (n,fp)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
fp is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
{ (Base_FinSeq (n,b₁)) where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= fp ) } is set
fp + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
{ (Base_FinSeq (n,b₁)) where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= fp + 1 ) } is set
h is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
h + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Base_FinSeq (n,(fp + 1)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( fp + 1 <= b₁ & b₁ <= n ) } is set
z is Element of bool the carrier of (TOP-REAL n)
id z is Relation-like z -defined z -valued Function-like one-to-one total quasi_total Element of bool [:z,z:]
[:z,z:] is Relation-like set
bool [:z,z:] is non empty set
k is set
gf is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Base_FinSeq (n,gf) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
k is Element of bool the carrier of (TOP-REAL n)
id k is Relation-like k -defined k -valued Function-like one-to-one total quasi_total Element of bool [:k,k:]
[:k,k:] is Relation-like set
bool [:k,k:] is non empty set
gf is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
gf * p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(gf * p) | k is Relation-like the carrier of (TOP-REAL n) -defined k -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( fp + 1 <= b₁ & b₁ <= n ) } /\ (Seg n) is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
{(fp + 1),n} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
card {(fp + 1),n} is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
m is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(gf * p) . m is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
card ( { b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( fp + 1 <= b₁ & b₁ <= n ) } /\ (Seg n)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
h is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
h . ((gf * p) . m) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(h . ((gf * p) . m)) . (fp + 1) is V11() real ext-real Element of REAL
h * gf is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
hg is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
hg * p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom (hg * p) is non empty set
(h * gf) * p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
c₂₂ is set
((h * gf) * p) . c₂₂ is Relation-like Function-like set
h . c₂₂ is Relation-like Function-like set
((gf * p) | k) . c₂₂ is Relation-like Function-like set
(gf * p) . c₂₂ is Relation-like Function-like set
i is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
H is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
i . H is V11() real ext-real Element of REAL
H is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
H is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Base_FinSeq (n,H) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
h * (gf * p) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(h * (gf * p)) . c₂₂ is Relation-like Function-like set
h . ((gf * p) . c₂₂) is Relation-like Function-like set
1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(hg * p) | z is Relation-like the carrier of (TOP-REAL n) -defined z -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
c₂₂ is set
((hg * p) | z) . c₂₂ is Relation-like Function-like set
(id z) . c₂₂ is set
h * (gf * p) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(hg * p) . c₂₂ is Relation-like Function-like set
i is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Base_FinSeq (n,i) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(h * (gf * p)) . c₂₂ is Relation-like Function-like set
len (h . ((gf * p) . m)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
H is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(h . ((gf * p) . m)) . H is V11() real ext-real Element of REAL
Base_FinSeq (n,H) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
0H is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
((h * gf) * p) . 0H is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
0H + (h . ((gf * p) . m)) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) is non empty Relation-like [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total Element of bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):]
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
the addF of (TOP-REAL n) . (0H,(h . ((gf * p) . m))) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
0H + (h . ((gf * p) . m)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
0H + m is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) . (0H,m) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
0H + m is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
((h * gf) * p) . (0H + m) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
len 0H is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
|.(((h * gf) * p) . (0H + m)).| is V11() real ext-real non negative Element of REAL
sqr (((h * gf) * p) . (0H + m)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (((h * gf) * p) . (0H + m))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (((h * gf) * p) . (0H + m)))) is V11() real ext-real Element of REAL
|.(0H + m).| is V11() real ext-real non negative Element of REAL
sqr (0H + m) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (0H + m)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (0H + m))) is V11() real ext-real Element of REAL
|.(0H + m).| ^2 is V11() real ext-real Element of REAL
|.(0H + m).| * |.(0H + m).| is V11() real ext-real non negative set
|.0H.| is V11() real ext-real non negative Element of REAL
sqr 0H is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr 0H) is V11() real ext-real Element of REAL
sqrt (Sum (sqr 0H)) is V11() real ext-real Element of REAL
|.0H.| ^2 is V11() real ext-real Element of REAL
|.0H.| * |.0H.| is V11() real ext-real non negative set
|((h . ((gf * p) . m)),0H)| is V11() real ext-real Element of REAL
2 * |((h . ((gf * p) . m)),0H)| is V11() real ext-real Element of REAL
(|.0H.| ^2) + (2 * |((h . ((gf * p) . m)),0H)|) is V11() real ext-real Element of REAL
|.(h . ((gf * p) . m)).| is V11() real ext-real non negative Element of REAL
sqr (h . ((gf * p) . m)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (h . ((gf * p) . m))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (h . ((gf * p) . m)))) is V11() real ext-real Element of REAL
|.(h . ((gf * p) . m)).| ^2 is V11() real ext-real Element of REAL
|.(h . ((gf * p) . m)).| * |.(h . ((gf * p) . m)).| is V11() real ext-real non negative set
((|.0H.| ^2) + (2 * |((h . ((gf * p) . m)),0H)|)) + (|.(h . ((gf * p) . m)).| ^2) is V11() real ext-real Element of REAL
0* n is Relation-like NAT -defined REAL -valued Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL n
REAL n is non empty functional FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of n -tuples_on REAL
(0* n) +* (H,1) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
len m is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
|(m,0H)| is V11() real ext-real Element of REAL
2 * |(m,0H)| is V11() real ext-real Element of REAL
(|.0H.| ^2) + (2 * |(m,0H)|) is V11() real ext-real Element of REAL
|.m.| is V11() real ext-real non negative Element of REAL
sqr m is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr m) is V11() real ext-real Element of REAL
sqrt (Sum (sqr m)) is V11() real ext-real Element of REAL
|.m.| ^2 is V11() real ext-real Element of REAL
|.m.| * |.m.| is V11() real ext-real non negative set
((|.0H.| ^2) + (2 * |(m,0H)|)) + (|.m.| ^2) is V11() real ext-real Element of REAL
m . H is V11() real ext-real Element of REAL
(m . H) * 1 is V11() real ext-real Element of REAL
((h . ((gf * p) . m)) . H) * 1 is V11() real ext-real Element of REAL
0* n is Relation-like NAT -defined REAL -valued Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of REAL n
REAL n is non empty functional FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of n -tuples_on REAL
(0* n) +* ((fp + 1),((h . ((gf * p) . m)) . (fp + 1))) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
len (0* n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
j is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(h . ((gf * p) . m)) . j is V11() real ext-real Element of REAL
j is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(h . ((gf * p) . m)) . j is V11() real ext-real Element of REAL
((0* n) +* ((fp + 1),((h . ((gf * p) . m)) . (fp + 1)))) . j is V11() real ext-real Element of REAL
dom (0* n) is V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
(0* n) . j is V11() real ext-real Element of REAL
len ((0* n) +* ((fp + 1),((h . ((gf * p) . m)) . (fp + 1)))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
|.((gf * p) . m).| is V11() real ext-real non negative Element of REAL
sqr ((gf * p) . m) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr ((gf * p) . m)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr ((gf * p) . m))) is V11() real ext-real Element of REAL
|.m.| is V11() real ext-real non negative Element of REAL
sqr m is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr m) is V11() real ext-real Element of REAL
sqrt (Sum (sqr m)) is V11() real ext-real Element of REAL
(0* n) +* ((fp + 1),1) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
abs 1 is V11() real ext-real Element of REAL
|.(h . ((gf * p) . m)).| is V11() real ext-real non negative Element of REAL
sqr (h . ((gf * p) . m)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (h . ((gf * p) . m))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (h . ((gf * p) . m)))) is V11() real ext-real Element of REAL
abs ((h . ((gf * p) . m)) . (fp + 1)) is V11() real ext-real Element of REAL
dom (id z) is set
dom ((hg * p) | z) is set
{ (Base_FinSeq (n,b₁)) where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
fp is Element of bool the carrier of (TOP-REAL n)
id fp is Relation-like fp -defined fp -valued Function-like one-to-one total quasi_total Element of bool [:fp,fp:]
[:fp,fp:] is Relation-like set
bool [:fp,fp:] is non empty set
z is set
fpz is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Base_FinSeq (n,fpz) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
z is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
z * p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(z * p) | fp is Relation-like the carrier of (TOP-REAL n) -defined fp -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
z is Element of bool the carrier of (TOP-REAL n)
id z is Relation-like z -defined z -valued Function-like one-to-one total quasi_total Element of bool [:z,z:]
[:z,z:] is Relation-like set
bool [:z,z:] is non empty set
fp is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
fp * p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(fp * p) | z is Relation-like the carrier of (TOP-REAL n) -defined z -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
fpz is Relation-like Function-like set
dom (fp * p) is non empty set
(fp * p) . fpz is Relation-like Function-like set
h is set
((fp * p) . fpz) . h is set
fpz . h is set
sq is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(fp * p) . sq is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
len ((fp * p) . sq) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom ((fp * p) . sq) is V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
((fp * p) . sq) . h is V11() real ext-real Element of REAL
len sq is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom sq is V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
f + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Base_FinSeq (n,z) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Lin z is non empty right_complementable strict V189() V190() V191() V192() V193() V194() V195() M25( TOP-REAL n)
((fp * p) . sq) . z is V11() real ext-real Element of REAL
sq . z is V11() real ext-real Element of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
Det p is V11() real ext-real Element of the carrier of F_Real
GFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions SubStr of GPFuncs the carrier of (TOP-REAL n)
GPFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions multMagma
the carrier of (GFuncs the carrier of (TOP-REAL n)) is non empty set
n1 is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Product n1 is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
dom n1 is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n1 | f is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg f is V16() V36() f -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
n1 | (Seg f) is Relation-like NAT -defined Seg f -defined NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSubsequence-like finite-support set
Product (n1 | f) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
f + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n1 | (f + 1) is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg (f + 1) is non empty V16() V36() f + 1 -element f + 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= f + 1 ) } is set
n1 | (Seg (f + 1)) is Relation-like NAT -defined Seg (f + 1) -defined NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSubsequence-like finite-support set
Product (n1 | (f + 1)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
z is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,z) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
width (n,z) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (n,z) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n1 . (f + 1) is set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
fpz is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
h is V11() real ext-real set
(n,h,z,fpz) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
Mx2Tran (n,h,z,fpz) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
z is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
<*z*> is non empty trivial Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like finite-support Function-yielding V235() V282() Element of 1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL n))
1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL n)) is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
[1,z] is set
{1,z} is non empty V36() set
{{1,z},{1}} is non empty V36() V40() set
{[1,z]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(n1 | f) ^ <*z*> is non empty Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
the carrier of (GFuncs the carrier of (TOP-REAL n)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
(Product (n1 | f)) * z is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
the multF of (GFuncs the carrier of (TOP-REAL n)) is non empty Relation-like [: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like total quasi_total Element of bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):]
[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
[:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty set
the multF of (GFuncs the carrier of (TOP-REAL n)) . ((Product (n1 | f)),z) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
(Mx2Tran (n,h,z,fpz)) * z is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
width (n,h,z,fpz) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (n,h,z,fpz) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran z is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Det z is V11() real ext-real Element of the carrier of F_Real
Mx2Tran (n,z) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran (n,h,z,fpz)) * (Mx2Tran (n,z)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,z) * (n,h,z,fpz) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran ((n,z) * (n,h,z,fpz)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Det (n,z) is V11() real ext-real Element of the carrier of F_Real
Det (n,h,z,fpz) is V11() real ext-real Element of the carrier of F_Real
(1. F_Real) * (1. F_Real) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
the multF of F_Real . ((1. F_Real),(1. F_Real)) is V11() real ext-real Element of the carrier of F_Real
K538((1. F_Real),(1. F_Real)) is V11() real ext-real Element of REAL
n1 | (len n1) is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg (len n1) is V16() V36() len n1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= len n1 ) } is set
n1 | (Seg (len n1)) is Relation-like NAT -defined Seg (len n1) -defined NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSubsequence-like finite-support set
n1 | {} is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg {} is empty ordinal natural V11() real ext-real non positive non negative V16() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
n1 | (Seg {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined Seg {} -defined NAT -defined RAT -valued the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
Product (n1 | {}) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
1. (F_Real,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran (1. (F_Real,n)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
<*> the carrier of (GFuncs the carrier of (TOP-REAL n)) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
the carrier of (GFuncs the carrier of (TOP-REAL n)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
1_ (GFuncs the carrier of (TOP-REAL n)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
the multF of (GFuncs the carrier of (TOP-REAL n)) is non empty Relation-like [: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like total quasi_total Element of bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):]
[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
[:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty set
the_unity_wrt the multF of (GFuncs the carrier of (TOP-REAL n)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
Det (1. (F_Real,n)) is V11() real ext-real Element of the carrier of F_Real
1_ F_Real is V11() real ext-real Element of the carrier of F_Real
f is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran f is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Det f is V11() real ext-real Element of the carrier of F_Real
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Det (n,p) is V11() real ext-real Element of the carrier of F_Real
Mx2Tran (n,p) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
len (n,p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
width (n,p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
{n} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
X is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
X * (Mx2Tran (n,p)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,X) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
width (n,X) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Mx2Tran (n,X) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,(X * (Mx2Tran (n,p)))) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
1. (F_Real,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(n,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
Det (n,X) is V11() real ext-real Element of the carrier of F_Real
(n,p) * (n,X) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Det ((n,p) * (n,X)) is V11() real ext-real Element of the carrier of F_Real
(1. F_Real) * (1. F_Real) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
the multF of F_Real . ((1. F_Real),(1. F_Real)) is V11() real ext-real Element of the carrier of F_Real
K538((1. F_Real),(1. F_Real)) is V11() real ext-real Element of REAL
rng (Mx2Tran (n,p)) is non empty set
rng X is non empty set
[#] (TOP-REAL n) is non empty non proper Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
dom X is non empty set
X /" is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom (X /") is non empty set
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Mx2Tran (1. (F_Real,n)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom (X * (Mx2Tran (n,p))) is non empty set
dom (id the carrier of (TOP-REAL n)) is non empty set
0. (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element V61( TOP-REAL n) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{(0. (TOP-REAL n))} is non empty trivial functional V36() V40() 1 -element set
rng (X * (Mx2Tran (n,p))) is non empty set
rng (id the carrier of (TOP-REAL n)) is non empty set
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
Mx2Tran (n,(X * (Mx2Tran (n,p)))) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Mx2Tran (n,n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Mx2Tran ((n,p) * (n,X)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(X /") * (X * (Mx2Tran (n,p))) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(X /") * X is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
((X /") * X) * (Mx2Tran (n,p)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(id the carrier of (TOP-REAL n)) * (Mx2Tran (n,p)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Det (n,p) is V11() real ext-real Element of the carrier of F_Real
Mx2Tran (n,p) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
len (n,p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
width (n,p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
1. (F_Real,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Det (1. (F_Real,n)) is V11() real ext-real Element of the carrier of F_Real
1_ F_Real is V11() real ext-real Element of the carrier of F_Real
{n} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
f is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
f * (Mx2Tran (n,p)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,f) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran (n,f) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Det (n,f) is V11() real ext-real Element of the carrier of F_Real
(n,p) * (n,f) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Det ((n,p) * (n,f)) is V11() real ext-real Element of the carrier of F_Real
(Det (n,p)) * (1. F_Real) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
the multF of F_Real . ((Det (n,p)),(1. F_Real)) is V11() real ext-real Element of the carrier of F_Real
K538((Det (n,p)),(1. F_Real)) is V11() real ext-real Element of REAL
width (n,f) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
Mx2Tran ((n,p) * (n,f)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,(f * (Mx2Tran (n,p)))) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(n,(f * (Mx2Tran (n,p)))) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(n,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
dom (f * (Mx2Tran (n,p))) is non empty set
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom (id the carrier of (TOP-REAL n)) is non empty set
0. (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element V61( TOP-REAL n) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{(0. (TOP-REAL n))} is non empty trivial functional V36() V40() 1 -element set
rng (f * (Mx2Tran (n,p))) is non empty set
rng (id the carrier of (TOP-REAL n)) is non empty set
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Mx2Tran (1. (F_Real,n)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
Det (n,n) is V11() real ext-real Element of the carrier of F_Real
(n,(f * (Mx2Tran (n,p)))) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(n,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
p is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL p is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL p) is non empty set
[: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):] is non empty set
Seg p is V16() V36() p -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
(p,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of p,p, the carrier of F_Real
q is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
(p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
Det (p,q) is V11() real ext-real Element of the carrier of F_Real
TR is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
(p,TR) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
q * TR is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued the carrier of (TOP-REAL p) -valued the carrier of (TOP-REAL p) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
Mx2Tran (p,n) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
Mx2Tran (p,q) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
(Mx2Tran (p,q)) * TR is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued the carrier of (TOP-REAL p) -valued the carrier of (TOP-REAL p) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
X is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
(p,X) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
Mx2Tran (p,X) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
width (p,q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (p,n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
width (p,n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(p,n) * (p,q) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of p,p, the carrier of F_Real
Mx2Tran ((p,n) * (p,q)) is non empty Relation-like the carrier of (TOP-REAL p) -defined the carrier of (TOP-REAL p) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL p), the carrier of (TOP-REAL p):]
Det (p,n) is V11() real ext-real Element of the carrier of F_Real
Det (p,X) is V11() real ext-real Element of the carrier of F_Real
(- (1. F_Real)) * (- (1. F_Real)) is V11() real ext-real Element of the carrier of F_Real
the multF of F_Real is non empty Relation-like [: the carrier of F_Real, the carrier of F_Real:] -defined the carrier of F_Real -valued Function-like total quasi_total complex-yielding ext-real-valued real-valued associative V287( the carrier of F_Real) Element of bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:]
[: the carrier of F_Real, the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
[:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty Relation-like complex-yielding ext-real-valued real-valued set
bool [:[: the carrier of F_Real, the carrier of F_Real:], the carrier of F_Real:] is non empty set
the multF of F_Real . ((- (1. F_Real)),(- (1. F_Real))) is V11() real ext-real Element of the carrier of F_Real
K538((- (1. F_Real)),(- (1. F_Real))) is V11() real ext-real Element of REAL
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
GFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions SubStr of GPFuncs the carrier of (TOP-REAL n)
GPFuncs the carrier of (TOP-REAL n) is non empty strict unital associative constituted-Functions multMagma
the carrier of (GFuncs the carrier of (TOP-REAL n)) is non empty set
n1 is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Product n1 is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
dom n1 is V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
Mx2Tran (n,p) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
len n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
f is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n1 | f is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg f is V16() V36() f -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= f ) } is set
n1 | (Seg f) is Relation-like NAT -defined Seg f -defined NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSubsequence-like finite-support set
Product (n1 | f) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
f + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n1 | (f + 1) is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg (f + 1) is non empty V16() V36() f + 1 -element f + 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= f + 1 ) } is set
n1 | (Seg (f + 1)) is Relation-like NAT -defined Seg (f + 1) -defined NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSubsequence-like finite-support set
Product (n1 | (f + 1)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
z is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,z) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
width (n,z) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (n,z) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n1 . (f + 1) is set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
fpz is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
h is V11() real ext-real set
(n,h,z,fpz) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
Mx2Tran (n,h,z,fpz) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
z is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
<*z*> is non empty trivial Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like constant V36() 1 -element FinSequence-like FinSubsequence-like finite-support Function-yielding V235() V282() Element of 1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL n))
1 -tuples_on the carrier of (GFuncs the carrier of (TOP-REAL n)) is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
[1,z] is set
{1,z} is non empty V36() set
{{1,z},{1}} is non empty V36() V40() set
{[1,z]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(n1 | f) ^ <*z*> is non empty Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
the carrier of (GFuncs the carrier of (TOP-REAL n)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
(Product (n1 | f)) * z is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
the multF of (GFuncs the carrier of (TOP-REAL n)) is non empty Relation-like [: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like total quasi_total Element of bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):]
[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
[:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty set
the multF of (GFuncs the carrier of (TOP-REAL n)) . ((Product (n1 | f)),z) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
(Mx2Tran (n,h,z,fpz)) * z is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
width (n,h,z,fpz) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (n,h,z,fpz) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran z is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Mx2Tran (n,z) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran (n,h,z,fpz)) * (Mx2Tran (n,z)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,z) * (n,h,z,fpz) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran ((n,z) * (n,h,z,fpz)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n1 | (len n1) is Relation-like NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg (len n1) is V16() V36() len n1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= len n1 ) } is set
n1 | (Seg (len n1)) is Relation-like NAT -defined Seg (len n1) -defined NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like V36() FinSubsequence-like finite-support set
n1 | {} is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() FinSequence of the carrier of (GFuncs the carrier of (TOP-REAL n))
Seg {} is empty ordinal natural V11() real ext-real non positive non negative V16() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
n1 | (Seg {}) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined Seg {} -defined NAT -defined RAT -valued the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
Product (n1 | {}) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
1. (F_Real,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran (1. (F_Real,n)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
<*> the carrier of (GFuncs the carrier of (TOP-REAL n)) is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of the carrier of (GFuncs the carrier of (TOP-REAL n)) *
the carrier of (GFuncs the carrier of (TOP-REAL n)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (GFuncs the carrier of (TOP-REAL n))
1_ (GFuncs the carrier of (TOP-REAL n)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
the multF of (GFuncs the carrier of (TOP-REAL n)) is non empty Relation-like [: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] -defined the carrier of (GFuncs the carrier of (TOP-REAL n)) -valued Function-like total quasi_total Element of bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):]
[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
[:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty Relation-like set
bool [:[: the carrier of (GFuncs the carrier of (TOP-REAL n)), the carrier of (GFuncs the carrier of (TOP-REAL n)):], the carrier of (GFuncs the carrier of (TOP-REAL n)):] is non empty set
the_unity_wrt the multF of (GFuncs the carrier of (TOP-REAL n)) is Relation-like Function-like Element of the carrier of (GFuncs the carrier of (TOP-REAL n))
f is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Mx2Tran f is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
p is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Det (n,p) is V11() real ext-real Element of the carrier of F_Real
Det (n,p) is V11() real ext-real Element of the carrier of F_Real
dom p is non empty set
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom (id the carrier of (TOP-REAL n)) is non empty set
0. (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element V61( TOP-REAL n) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{(0. (TOP-REAL n))} is non empty trivial functional V36() V40() 1 -element set
rng p is non empty set
rng (id the carrier of (TOP-REAL n)) is non empty set
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
(n,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
Mx2Tran (n,n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,(Mx2Tran (n,n))) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
p * (Mx2Tran (n,n)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,n) ~ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(n,n) @ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(n,(p * (Mx2Tran (n,n)))) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(n,(Mx2Tran (n,n))) * (n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(n,n) * (n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
((n,n) ~) * (n,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
1. (F_Real,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
width ((n,n) ~) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (n,n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
width (n,n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (n,p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
((n,n) ~) * ((n,n) * (n,p)) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
(((n,n) ~) * (n,n)) * (n,p) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Det (n,p) is V11() real ext-real Element of the carrier of F_Real
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
TR is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n1 is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(n,n1) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
Det (n,n1) is V11() real ext-real Element of the carrier of F_Real
Det (n,n1) is V11() real ext-real Element of the carrier of F_Real
dom TR is non empty set
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom (id the carrier of (TOP-REAL n)) is non empty set
0. (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element V61( TOP-REAL n) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
{(0. (TOP-REAL n))} is non empty trivial functional V36() V40() 1 -element set
rng TR is non empty set
rng (id the carrier of (TOP-REAL n)) is non empty set
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
dom TR is non empty set
(n,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
Mx2Tran (n,n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran (n,n)) /" is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
rng (Mx2Tran (n,n)) is non empty set
[#] (TOP-REAL n) is non empty non proper Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
(n,(Mx2Tran (n,n))) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
TR * (Mx2Tran (n,n)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(TR * (Mx2Tran (n,n))) * ((Mx2Tran (n,n)) /") is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran (n,n)) * ((Mx2Tran (n,n)) /") is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
TR * ((Mx2Tran (n,n)) * ((Mx2Tran (n,n)) /")) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
TR * (id the carrier of (TOP-REAL n)) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Det (n,n1) is V11() real ext-real Element of the carrier of F_Real
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
(n,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
1. (F_Real,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of n,n, the carrier of F_Real
p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
|.p.| is V11() real ext-real non negative Element of REAL
sqr p is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr p) is V11() real ext-real Element of REAL
sqrt (Sum (sqr p)) is V11() real ext-real Element of REAL
q is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
|.q.| is V11() real ext-real non negative Element of REAL
sqr q is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr q) is V11() real ext-real Element of REAL
sqrt (Sum (sqr q)) is V11() real ext-real Element of REAL
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n1 is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n1 . p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(n,n1) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Orthogonal Matrix of n,n, the carrier of F_Real
TOP-REAL 1 is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL 1) is non empty set
1. (F_Real,1) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Matrix of 1,1, the carrier of F_Real
Mx2Tran (1. (F_Real,1)) is non empty Relation-like the carrier of (TOP-REAL 1) -defined the carrier of (TOP-REAL 1) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL 1), the carrier of (TOP-REAL 1):]
[: the carrier of (TOP-REAL 1), the carrier of (TOP-REAL 1):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL 1), the carrier of (TOP-REAL 1):] is non empty set
len p is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
p . 1 is V11() real ext-real Element of REAL
<*(p . 1)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
[1,(p . 1)] is set
{1,(p . 1)} is non empty V36() V155() V156() V157() set
{{1,(p . 1)},{1}} is non empty V36() V40() set
{[1,(p . 1)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
(1,1) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of 1,1, the carrier of F_Real
Mx2Tran (1,1) is non empty Relation-like the carrier of (TOP-REAL 1) -defined the carrier of (TOP-REAL 1) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL 1), the carrier of (TOP-REAL 1):]
TOP-REAL 1 is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL 1) is non empty set
[: the carrier of (TOP-REAL 1), the carrier of (TOP-REAL 1):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL 1), the carrier of (TOP-REAL 1):] is non empty set
n1 is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n1 . p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(n,n1) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular Orthogonal Matrix of n,n, the carrier of F_Real
q . 1 is V11() real ext-real Element of REAL
(q . 1) ^2 is V11() real ext-real Element of REAL
(q . 1) * (q . 1) is V11() real ext-real set
(p . 1) ^2 is V11() real ext-real Element of REAL
(p . 1) * (p . 1) is V11() real ext-real set
sqr <*(p . 1)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
Sum (sqr <*(p . 1)*>) is V11() real ext-real Element of REAL
sqrt (Sum (sqr <*(p . 1)*>)) is V11() real ext-real Element of REAL
<*((p . 1) ^2)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,((p . 1) ^2)] is set
{1,((p . 1) ^2)} is non empty V36() V155() V156() V157() set
{{1,((p . 1) ^2)},{1}} is non empty V36() V40() set
{[1,((p . 1) ^2)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
Sum <*((p . 1) ^2)*> is V11() real ext-real Element of REAL
sqrt (Sum <*((p . 1) ^2)*>) is V11() real ext-real Element of REAL
sqrt ((p . 1) ^2) is V11() real ext-real Element of REAL
len q is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
<*(q . 1)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(q . 1)] is set
{1,(q . 1)} is non empty V36() V155() V156() V157() set
{{1,(q . 1)},{1}} is non empty V36() V40() set
{[1,(q . 1)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
sqr <*(q . 1)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
Sum (sqr <*(q . 1)*>) is V11() real ext-real Element of REAL
sqrt (Sum (sqr <*(q . 1)*>)) is V11() real ext-real Element of REAL
<*((q . 1) ^2)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,((q . 1) ^2)] is set
{1,((q . 1) ^2)} is non empty V36() V155() V156() V157() set
{{1,((q . 1) ^2)},{1}} is non empty V36() V40() set
{[1,((q . 1) ^2)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
Sum <*((q . 1) ^2)*> is V11() real ext-real Element of REAL
sqrt (Sum <*((q . 1) ^2)*>) is V11() real ext-real Element of REAL
sqrt ((q . 1) ^2) is V11() real ext-real Element of REAL
len (n1 . p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(n1 . p) . 1 is V11() real ext-real Element of REAL
<*((n1 . p) . 1)*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,((n1 . p) . 1)] is set
{1,((n1 . p) . 1)} is non empty V36() V155() V156() V157() set
{{1,((n1 . p) . 1)},{1}} is non empty V36() V40() set
{[1,((n1 . p) . 1)]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
- (p . 1) is V11() real ext-real Element of REAL
<*(- (p . 1))*> is non empty trivial Relation-like NAT -defined REAL -valued Function-like one-to-one constant V36() 1 -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued V149() decreasing non-decreasing non-increasing finite-support Element of 1 -tuples_on REAL
[1,(- (p . 1))] is set
{1,(- (p . 1))} is non empty V36() V155() V156() V157() set
{{1,(- (p . 1))},{1}} is non empty V36() V40() set
{[1,(- (p . 1))]} is non empty trivial Relation-like Function-like constant V36() 1 -element finite-support set
n is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
TOP-REAL n is non empty TopSpace-like right_complementable constituted-Functions constituted-FinSeqs V189() V190() V191() V192() V193() V194() V195() V228() L16()
the carrier of (TOP-REAL n) is non empty set
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty Relation-like set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
|.p.| is V11() real ext-real non negative Element of REAL
sqr p is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr p) is V11() real ext-real Element of REAL
sqrt (Sum (sqr p)) is V11() real ext-real Element of REAL
q is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
|.q.| is V11() real ext-real non negative Element of REAL
sqr q is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr q) is V11() real ext-real Element of REAL
sqrt (Sum (sqr q)) is V11() real ext-real Element of REAL
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n1 is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
n1 . p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
0. (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element V61( TOP-REAL n) FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
Seg n is V16() V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
{ b₁ where b₁ is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
card (Seg n) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(n1 + 1) + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(n1 + 1) + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
z is set
len z is ordinal cardinal set
X is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
X . p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
z is set
len z is ordinal cardinal set
X is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
X . p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
sqr (X . p) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
(Seg n) \ z is V16() V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
h is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(sqr q) . h is V11() real ext-real Element of REAL
(sqr (X . p)) . h is V11() real ext-real Element of REAL
h is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(sqr q) . h is V11() real ext-real Element of REAL
(sqr (X . p)) . h is V11() real ext-real Element of REAL
{h} is non empty trivial V36() V40() 1 -element V155() V156() V157() V158() V159() V160() Element of bool NAT
z \/ {h} is non empty set
(X . p) . h is V11() real ext-real Element of REAL
((X . p) . h) ^2 is V11() real ext-real Element of REAL
((X . p) . h) * ((X . p) . h) is V11() real ext-real set
q . h is V11() real ext-real Element of REAL
(q . h) ^2 is V11() real ext-real Element of REAL
(q . h) * (q . h) is V11() real ext-real set
len (z \/ {h}) is non empty ordinal cardinal set
z is set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(X . p) . z is V11() real ext-real Element of REAL
((X . p) . z) ^2 is V11() real ext-real Element of REAL
((X . p) . z) * ((X . p) . z) is V11() real ext-real set
{} + ((q . h) ^2) is V11() real ext-real Element of REAL
(((X . p) . z) ^2) + (((X . p) . h) ^2) is V11() real ext-real Element of REAL
k is V11() real ext-real set
(n,k,z,h) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
Mx2Tran (n,k,z,h) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran (n,k,z,h)) . (X . p) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
((Mx2Tran (n,k,z,h)) . (X . p)) . h is V11() real ext-real Element of REAL
{h,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
k is V11() real ext-real set
(n,k,h,z) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like V36() FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Function-yielding V235() tabular invertible Matrix of n,n, the carrier of F_Real
Mx2Tran (n,k,h,z) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
(Mx2Tran (n,k,h,z)) . (X . p) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
((Mx2Tran (n,k,h,z)) . (X . p)) . h is V11() real ext-real Element of REAL
{z,h} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{h,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{h,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
{h,z} is non empty V36() V40() V155() V156() V157() V158() V159() V160() Element of bool NAT
k is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
k . (X . p) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(k . (X . p)) . h is V11() real ext-real Element of REAL
k is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
k . (X . p) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(k . (X . p)) . h is V11() real ext-real Element of REAL
k * X is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
gf is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
gf . p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
m is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(gf . p) . m is V11() real ext-real Element of REAL
q . m is V11() real ext-real Element of REAL
dom gf is non empty set
dom k is non empty set
(k . (X . p)) . m is V11() real ext-real Element of REAL
(X . p) . m is V11() real ext-real Element of REAL
(Seg n) \ z is V16() V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
Sum (sqr (X . p)) is V11() real ext-real Element of REAL
|.(X . p).| is V11() real ext-real non negative Element of REAL
sqrt (Sum (sqr (X . p))) is V11() real ext-real Element of REAL
@ (X . p) is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
@ (@ (X . p)) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
len (sqr (X . p)) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len (X . p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
@ q is Relation-like NAT -defined the carrier of F_Real -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of the carrier of F_Real
@ (@ q) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
len (sqr q) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len q is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
z is set
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
h is Relation-like NAT -defined REAL -valued Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of n -tuples_on REAL
sq is Relation-like NAT -defined REAL -valued Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of n -tuples_on REAL
k is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
h . k is V11() real ext-real Element of REAL
sq . k is V11() real ext-real Element of REAL
(X . p) . k is V11() real ext-real Element of REAL
q . k is V11() real ext-real Element of REAL
(q . k) ^2 is V11() real ext-real Element of REAL
(q . k) * (q . k) is V11() real ext-real set
sq . z is V11() real ext-real Element of REAL
h . z is V11() real ext-real Element of REAL
(Seg n) \ z is V16() V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
h is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(sqr q) . h is V11() real ext-real Element of REAL
(sqr (X . p)) . h is V11() real ext-real Element of REAL
n - 1 is V11() real ext-real V85() Element of REAL
n + (- 1) is V11() real ext-real V85() set
n1 is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
n1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
{} + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
id (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total quasi_total quasi_total additive additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
id the carrier of (TOP-REAL n) is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one total quasi_total quasi_total FinSequence-yielding Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
f is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
f . p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
X is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
len X is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() set
card X is empty ordinal natural V11() real ext-real non positive non negative Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V36() V37() V40() cardinal {} -element V85() V86() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding ext-real-valued real-valued natural-valued V155() V156() V157() V158() V159() V160() V161() FinSequence-yielding finite-support Function-yielding V235() V282() Element of omega
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
(f . p) . z is V11() real ext-real Element of REAL
q . z is V11() real ext-real Element of REAL
X is set
len X is ordinal cardinal set
f is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() being_homeomorphism (n) (n) Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
f . p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
(Seg n) \ X is V16() V36() V155() V156() V157() V158() V159() V160() Element of bool NAT
card ((Seg n) \ X) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of omega
n - n1 is V11() real ext-real V85() set
- n1 is V11() real ext-real non positive V85() set
n + (- n1) is V11() real ext-real V85() set
z is set
{z} is non empty trivial V36() 1 -element set
sqr (f . p) is Relation-like NAT -defined REAL -valued Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support FinSequence of REAL
Sum (sqr (f . p)) is V11() real ext-real Element of REAL
|.(f . p).| is V11() real ext-real non negative Element of REAL
sqrt (Sum (sqr (f . p))) is V11() real ext-real Element of REAL
z is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
q . z is V11() real ext-real Element of REAL
(f . p) +* (z,(q . z)) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
len (f . p) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
dom (f . p) is V36() n -element V155() V156() V157() V158() V159() V160() Element of bool NAT
h is ordinal natural V11() real ext-real non negative V36() cardinal V85() set
((f . p) +* (z,(q . z))) . h is V11() real ext-real Element of REAL
q . h is V11() real ext-real Element of REAL
(f . p) . h is V11() real ext-real Element of REAL
len ((f . p) +* (z,(q . z))) is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
len q is ordinal natural V11() real ext-real non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
(f . p) . z is V11() real ext-real Element of REAL
((f . p) . z) ^2 is V11() real ext-real Element of REAL
((f . p) . z) * ((f . p) . z) is V11() real ext-real set
(Sum (sqr (f . p))) - (((f . p) . z) ^2) is V11() real ext-real Element of REAL
- (((f . p) . z) ^2) is V11() real ext-real set
(Sum (sqr (f . p))) + (- (((f . p) . z) ^2)) is V11() real ext-real set
(q . z) ^2 is V11() real ext-real Element of REAL
(q . z) * (q . z) is V11() real ext-real set
((Sum (sqr (f . p))) - (((f . p) . z) ^2)) + ((q . z) ^2) is V11() real ext-real Element of REAL
- (q . z) is V11() real ext-real Element of REAL
1 + 1 is non empty ordinal natural V11() real ext-real positive non negative V36() cardinal V85() V86() V155() V156() V157() V158() V159() V160() Element of NAT
- ((f . p) . z) is V11() real ext-real Element of REAL
(f . p) +* (z,(- ((f . p) . z))) is Relation-like NAT -defined Function-like V36() FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support set
h is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
h . (f . p) is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
h * f is non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued the carrier of (TOP-REAL n) -valued Function-like total total total quasi_total quasi_total quasi_total additive FinSequence-yielding homogeneous Function-yielding V235() Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
dom (h * f) is non empty set
(h * f) . p is Relation-like NAT -defined Function-like V36() n -element FinSequence-like FinSubsequence-like complex-yielding ext-real-valued real-valued finite-support Element of the carrier of (TOP-REAL n)
- (q . z) is V11() real ext-real Element of REAL